Carlos Alberto Conceição António

Gonçalo das Neves Carneiro

English

Repositório Aberto da Universidade do Porto

    Optimization of Mechanical Systems (Otimização de Sistemas Mecânicos) M.EM005  Optimal Design Applications         Carlos Alberto da Conceição António Gonçalo Maria Pereira das Neves Carneiro    Optimization of Mechanical Systems / Optimal Design Applications Carlos Conceição António / Gonçalo das Neves Carneiro  2   Optimization of Mechanical Systems / Optimal Design Applications Carlos Conceição António / Gonçalo das Neves Carneiro  3 Contents   1. Polynomial problems 1.1 Unconstrained problems 1.2 Test functions 1.3 Constrained optimization problems   2. Mechanical design problems 2.1 Structural and mechanical applications 2.2 Thermal and fluid mechanics applications    Optimization of Mechanical Systems / Optimal Design Applications Carlos Conceição António / Gonçalo das Neves Carneiro  4 Preface  Optimization is a branch of Mathematics and Numerical Methods that has been the subject of intensive theoretical and applied research over the last decades. In particular, Engineering is one of the areas where optimization has found propitious ground for its application and theoretical and practical development. The development of numerical methods in engineering and science in general, as well as the parallel development of computational capacities, has made available to engineers and scientists powerful tools for the analysis and resolution of the most varied and complex problems in their respective areas. Once equipped with the ability to analyse and solve complex problems, the natural path of the engineer and scientist is to try to obtain the best solution for these problems, the optimal solution. Design optimization can be defined as the rational establishment of a design that is the best within all possible designs according to one or more predefined objectives and obeying a prescribed set of geometric constraints and/or constraints related to the behaviour and integrity of systems in engineering, technological constraints, etc. The correct formulation of the design optimization problem is critical because the quality of the optimal solution is dependent on the formulation of the problem. For example, if a critical constraint is omitted in the formulation, the optimal solution will probably not satisfy the constraint in question. Moreover, if too many constraints are considered or if they are inconsistent, there may be no solution to the problem. However, once the problem is formulated properly, there is usually an algorithm to solve it. To apply mathematical concepts to optimal design research, the underlying mathematical problem must be formulated and the respective model built. To do so, it is necessary to define the design variables, the objectives and the constraints of the problem. The nature of these mathematical entities depends on the available information about the design problem, which can be deterministic, probabilistic/stochastic or fuzzy. On the other hand, optimization algorithms can be classified in three major classes: mathematical programming, optimality criteria and bio-inspired algorithms.  This document presents a set of applications that represent a challenge for students in learning the curricular unit of Optimization of Mechanical Systems. Initially are suggested examples of unconstrained problems involving the calculation of the extreme values of polynomial functions. It is also proposed the minimization/maximization of polynomial functions subject to constraints. Finally, real scenario problems involving the optimal design of mechanical systems in two areas are proposed: i) structures and mechanical components; ii) thermal and fluid mechanics systems. Students are invited to use optimization methods with different approaches in the search for extreme values, comparing the results obtained and the associated costs.  FEUP, 21 September 2021  Carlos Conceição António, Prof. Catedrático Gonçalo das Neves Carneiro, Prof. Auxiliar Optimization of Mechanical Systems / Optimal Design Applications Carlos Conceição António / Gonçalo das Neves Carneiro  5 1. Polynomial problems (PP) 1.1 Unconstrained problems Exercise PP 1:  Minimize  !("#, "$) = "# % "$ + 2"#$ + 2"#"$ + "$$                          (1.1) from the starting point  &'("#, "$) = (0,0)  using the steepest descent (Cauchy) method.  Exercise PP 2:  Solve the following problem using the steepest descent (Cauchy) method: Minimize  !("#, "$) = "#$ + "$$ + "#"$ % 1                                       (1.2)  Exercise PP 3:  Consider the following problem: Maximize  !("#, "$) = "# % "$ % 2"#"$ % "#$ % 2"$$                            (1.3) a) Write the necessary optimality conditions. b) Determine the optimum using the Fletcher and Reeves conjugate gradient method. c) Determine the optimum using the Polack and Ribière conjugate gradient method.  Exercise PP 4:  Minimize  !("#, "$) = "# % "$ + 2"#$ + 2"#"$ + "$$                             (1.4) from the starting point  &'("#, "$) = (0,0)  using the DFP and BFGS methods [3].    Optimization of Mechanical Systems / Optimal Design Applications Carlos Conceição António / Gonçalo das Neves Carneiro  6 Exercise PP 5:  Use the steepest descent method to find the minimum of the unconstrained objective function *(", -) given by: *(", -)2" + $./3 + 34                                                      (1.5) Solve the problem analytically and compare with the numerical results. Analyse the performance of the method.  Exercise PP 6:  Minimize the function: !(", -, 5) = "- + #/6% 17-$ + 5                                            (1.6) As the initial point consider the point with coordinates (180, 089, 089), in the search space. Choose two different unconstrained search methods and two different univariate search methods, to determine the optimum search-step. Combine the methods and compare their performance.  1.2 Test functions  Exercise PP 7:  The use of test functions aims to show how well a specific algorithm works compared to other algorithms. Each test function is minimized from a standard starting point. The total number of function evaluations required to find the optimum solution is taken as a measure of the efficiency of the algorithm. Some of the commonly used test functions are given below. The initial solution is denoted by &' and the optimal solution is represented by &:. 1. Rosenbrock’s parabolic valley [1]: !("#, "$) = 100;("$ % "#$)$ + (1 % "#)$                                   (1.7) &'("#, "$) = (%182, 180) ,   !(&') = 2<80 &: = (180, 180) ,   !(&:) = 080  Optimization of Mechanical Systems / Optimal Design Applications Carlos Conceição António / Gonçalo das Neves Carneiro  7 2. A quadratic function [1]: !("#, "$) = ("# + 2"$ % >)$ + (2"# + "$ % 9)$                                   (1.8) &'("#, "$) = (0, 0) ,   !(&') = ><80 &: = (1, ?) ,   !(&:) = 080  3. Powell’s quartic function [1]: !("#, "$, "4, "@) = ("# + 10"$)$ + 9;("4 % "@)$ +("$ % 2"4)@ + 10;("# % "@)@                        (1.9) &'("#, "$, "4, "@) = (?,%1, 0, 1) ,   !(&') = 21980 &: = (0, 0, 0, 0) ,   !(&:) = 080  4. Fletcher and Powell’s helical valley [1]: !("#, "$, "4) = 100; A["4 % 10;B("#, "$)]$ + CD"#$ + "$$ % 1E$F + "4$         (1.10) where  2GB("#, "$) =HIJIK LMNOLP Q"$"#R ;;;;;S!;;;"# T 0G + LMNOLP Q"$"#R ;;S!;;;"# U 0 &'("#, "$, "4) = (%1, 0, 0) ,   !(&') = 1000080 &: = (1, 0, 0) ,   !(&:) = 080  5. A nonlinear function of three variables [1]: !("#, "$, "4) = ##V(/WX/Y)Y + ZSP \#$G;"$;"4^ + _"` a%\/WV/b/Y % 2^$c           (1.11) &'("#, "$, "4) = (0, 1, 2) ,   !(&') = 185 &: = (1, 1, 1) ,   !(&:) = ?80   (maximum)  Optimization of Mechanical Systems / Optimal Design Applications Carlos Conceição António / Gonçalo das Neves Carneiro  8 6. Freudenstein and Roth function [1]: !("#, "$) = ; {%1? + "# + [(9 % "$);"$ % 2];"$}$ +{%2d + "# + [("$ + 1);"$ % 1<];"$}$                            (1.12) &'("#, "$) = (089, %2) ,   !(&') = <0085 &: = (9, <) ,   !(&:) = 080 &:: = (118<1,%08ed7e) ,   !(&::) = <e8de<2  7. Beale’s function [1]: !("#, "$) = ; [189 % "#;(1 % "$);]$ + [2829 % "#;(1 % "$$);]$ +[28729 % "#;(1 % "$4);]$                                                             (1.13) &'("#, "$) = (1, 1) ,   !(&') = 1<820?129 &: = (?, 089) ,   !(&:) = 080  8. Wood’s function [1]: !("#, "$, "4, "@) = [10;("$ % "#$)]$ + (1 % "#)$ + d0;("@ % "4$)$ +(1 % "4)$ + 10;("$ + "@ % 2)$ + 081;("$ % "@) &'("#, "$, "4, "@) = (%?,%1,%?, %1) ,   !(&') = 1d1d280 &: = (1, 1, 1, 1) ,   !(&:) = 080     (1.14) 1.3 Constrained optimization problems Exercise PP 8:  Consider the following problem: Minimize !("#, "$) = 2("# % 2)$ + <("$ % 1)$ subject to 2"# + e"$ f 7 2"# g 2"$                                                        (1.15) 

Optimization of mechanical systems - optimal design applications

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Optimization of mechanical systems - optimal design applications

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