Structural Optimization Design ( Structural Analysis & Optimization ) Hai Huang School of Astronautics, BUAA
Mai functions of Structures and mechanisms in a spacecraft Preserving the outline configuration of the spacecraft; Providing the space to install or mount other subsystems or equipments, meanwhile guaranteeing the precision of the installation; supporting and protecting equipments, and ensuring their safety under various loading cases; providing enough stiffness, strength as well as thermal protection, and ensuring the integrality of the spacecraft; And supplying other functions such as unlocking, deploying, releasing, separation, pointing and locking of solar array or antenna. 2 2019年2月26日 2019年2月26日
Traditional Structural Design Question: Best design? Could be further improved? Structural optimization (Structural Synthesis, Structural Optimization Design, Structural Design Optimization) —— a kind of design technology to find the best solutions for structure systems to be designed —> Structural Analysis —> Design Adjust Initial Design —> Re-Analysis —>Re-Adjust … —> Until Satisfying 2019年2月26日
Chapter 1 Introduction to Structural Optimization §1.1 General concepts of Structural Optimization Optimization —— generally refers to find the best solution for projects, designs, products etc. 4 Examples (1) Bus routes (2) Area of rectangle (3) Beam sectional function (4) Engineering design 2019年2月26日
(1) Bus routes 2019年2月26日
(2) Area of rectangle 2019年2月26日
Fig. 1.3 Thin walled Structural Design (3) Engineering design Fig. 1.3 Thin walled Structural Design 2019年2月26日
§1.2 Mathematical Basis of Optimization 1.2.1 Mathematical Programming Find —Design Variables —Objective Function (NLP) —Constraints —Constraint functions 2019年2月26日
(a) convex set (b) non-convex set , 1.2.2 Definitions in optimization Convex Set A set is said to be convex if, for arbitrary two points , , in S, for (a) convex set (b) non-convex set 2019年2月26日
(a) convex-like function (b) non convex-like function Convex-like Functions (a) convex-like function (b) non convex-like function Convex –like Function 2019年2月26日
Convex Set ---- feasible domain Convex Programming Convex Set ---- feasible domain Convex-like Function---- objective function 2019年2月26日
Structural Analysis and Optimization Quick Review
Theorem 1 Suppose and are differentiable, and is the optimum solution of (NLP), at which the constraints meet the eligibility requirement, then it must exist a set of parameters called Lagrange multipliers which let Kuhn-Tucker (KT) condition be satisfied. 2019年2月26日
Kuhn-Tucker conditions If is the optimum of (NLP), and at the point meet the eligibility requirement, then 2019年2月26日
The gradients of objective and constraint functions at
Theorem 3 If problem (NLP) is a convex programming, then any local optimum of the problem must be a global minimum; furthermore if and only if the function is strictly convex on the convex feasible domain R, the minimum of the problem (NLP) is unique. Theorem 4 If problem (NLP) is a convex programming, and satisfies Kuhn- Tucker condition (KT), then must be the optimum of problem (NLP). 2019年2月26日
Sufficient conditions to meet eligibility 2019年2月26日
Traditional Structural Design Structural optimization (Structural Synthesis, Structural Optimization Design, Structural Design Optimization) ——Combining the theories of mathematical programming and the methods of structural analysis to seek the best feasible design according to a selected quantitative measure of effectiveness for loaded structures with computers as tools. —> Structural Analysis —> Design Adjust Initial Design —> Re-Analysis —>Re-Adjust … —> Until Satisfying 2019年2月26日
§1.3 Model of typical structural optimization Optimum objective —structural weight Constraint conditions —strength, deformation, dynamic (vibration) performance, etc. Design variables (1) Cross-sectional Areas (2) Shape variables (3) Topology variables 2019年2月26日
3 Kinds of Structural Optimization 1。 Cross-sectional Areas 10 Bar truss 2019年2月26日
3 Kinds of Structural Optimization 2。 Shape variables 15 Bar truss 2019年2月26日
3 Kinds of Structural Optimization 3。 Topology variables Original After Optimization 2019年2月26日
Model of typical structural optimization is a mathematical programming Characteristics 1。 implicit function 2。Large number of variables 3。 Large number of constraints 2019年2月26日
Characteristics of Typical Structural Optimization (1) Constraint function usually is a complex non-linearly implicit function of design variables. Only numerical solutions are available using structural analysis such as FEM. 2019年2月26日
(2) Design variable space is usually very high dimensional. 2019年2月26日
Solution with Computer as A Tool (3) Automatic design produces by using computers and develops with computer technology Numerical Optimization Computer Code + FEM ==》 Solution with Computer as A Tool 2019年2月26日
E-3 Navigation Satellite Compacted state Deployed state 2019年2月26日
History of structural optimization (1) in 1960, L.A.Scmmit introduced the theories of mathematical programming into structural design (2) Criterion methods (60s~70s) (3) Approximation concepts ( 1976) (4) Programming methods (after 1976) (5) Dual method (proposed in 1980) (6) Applications and Topology research 2019年2月26日
Concerned research areas Efficient numerical algorithm Adaptive ability of the methods Practical engineering problem applications Sensitivities analysis Shape and topology optimization. 2019年2月26日
Typical topics the course will conduct (1) Criterion methods (2) Sensitivity analysis Calculating for derivatives of primal constraint functions respect to design variables. (3) Approximation concepts and programming methods (4) Dual methods (5) Two-level multi-point Approximation method 2019年2月26日
Thank You for your attention ! 2019年2月26日