1. 1 汽车优化设计 王琥 wanghu@hnu.edu.cn 湖南大学 机械与运载工程学院 汽车车身先进设计制造国家重点实验室

2. 2 什么是优化 Definition – In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories depending on whether the variables are continuous or discrete. – 更为通俗的表达 “Making things better” “Generating more profit” “Determining the best” “Do more with less” Papalambros: “The determination of values for design variables which minimize (maximize) the objective, while satisfying all constraints”

3. 3 优化的历史 Ancient Greek philosophers: geometrical optimization problems Zenodorus, 200 B.C.: “A sphere encloses the greatest volume for a given surface area” 《论等积形》 (On isometric figures) ? g ● Newton, Leibniz, Bernoulli, Del’Hospital (1697): “Brachistochrone Problem”: 最速降线问题

4. 4 里程碑 Lagrange (1750): constrained minimization Cauchy (1847): steepest descent Dantzig (1947): Simplex method (LP) Kuhn, Tucker (1951): optimality conditions Karmakar (1984): interior point method (LP) Bendsoe, Kikuchi (1988): topology optimization

5. 5 优化所带来的 … 优化的主要应用 – Getting a design/system to work – Reaching the optimal performance – Making a design/system reliable and robust 优化能帮助我们了解什么 – Design problem – Underlying physics – Model weaknesses

6. 6 优化问题的数学表达 优化问题的三要素 – Design variables （设计变量） : variables with which the design problem is parameterized: – Objective: quantity that is to be minimized (maximized) Usually denoted by: ( “cost function”) – Constraint: condition that has to be satisfied Inequality constraint: Equality constraint:

7. 7 优化问题的数学表达 优化问题最终的数学表达式

8. 8 目前优化方法的瓶颈 维数灾难 Curse of dimensionality Looks complicated … why not just sample the design space, and take the best one? – Consider problem with n design variables – Sample each variable with m samples – Number of computations required: m n 并行计算 – 并行机制 – 大量的计算所带来的资源占用 Take 1 s per computation, 10 variables, 10 samples: total time 317 years!

9. 9 优化的基本流程 Optimization-based design process: Collect data to describe the system Estimate initial design Analyze the system Check the constraints Does the design satisfy convergence criteria? Change the design using an optimization method Done Identify: 1.Design variables 2.Objective function 3.Constraints

10. 10 当代优化方法的进展 优化技术变得越来越不可或缺 – Increasing availability of numerical modeling techniques – Increasing availability of cheap computer power – Increased competition, global markets – Better and more powerful optimization techniques – Increasingly expensive production processes (trial-and-error approach too expensive) – More engineers having optimization knowledge

11. 11 优化设计方法存在的问题 主要瓶颈 – Proper problem formulation critical! – Choosing the right algorithm for a given problem – Many algorithms contain lots of control parameters – Optimization tends to exploit weaknesses in models – Optimization can result in very sensitive designs – Some problems are simply too hard / large / expensive

12. 12 优化的分类 尺寸优化，材料优化 形状优化 拓扑优化 Yamaha R1

13. 13    优化应用 Car shape optimization for fuel consumption reductionCar shape optimization for fuel consumption reduction Aerodynamic drag 74% Others 26% Fuel consumption repartition for a car at 120 km/h: Possibility of action to reduce consumption : - Motors evolution - Weight reduction - Car shape optimization Aerodynamic flow at the rear of Peugeot 206

14. 14 优化应用 C x_initial =0.21949 iso-surfaces of pressure C x_final =0.11725 (flow lines coloured by the longitudinal speed)

15. 15 国际工程优化领域的主要问题 高维问题 (Curse of dimensionality) The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces (often with hundreds or thousands of dimensions) that do not occur in low- dimensional settings such as the three-dimensional physical space of everyday experience.

16. 16 国际工程优化领域的主要问题 CAD/CAE 的无缝连接

17. 17 国际工程优化领域的主要问题 日益复杂的有限元模型 – 快速计算方法