1 / 25

Example II: Linear truss structure

Example II: Linear truss structure. Optimization goal is to minimize the mass of the structure Cross section areas of trusses as design variables Maximum stress in each element as inequality constraints Maximum displacement in loading points as inequality constraints

johnna
Download Presentation

Example II: Linear truss structure

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Example II: Linear truss structure • Optimization goal is to minimize the mass of the structure • Cross section areas of trusses as design variables • Maximum stress in each element as inequality constraints • Maximum displacement in loading points as inequality constraints • Gradient-based and ARSM optimization perform much better if constraint equations are formulated separately instead of using total max_stress and max_disp as constraints Part 4: Multidisciplinary Optimization

  2. Example II: Sensitivity analysis • MOP indicates only a1, a3, a8 as important variables for maximum stress and displacements,but all inputs are important for objective function Part 4: Multidisciplinary Optimization

  3. Example II: Sensitivity analysis a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 max_stress max_disp stress10 stress9 stress8 stress8 stress6 stress5 stress4 stress3 stress2 stress1 disp4 disp2 mass MOP filter • For single stress values used in constraint equations, each input variable occurs at least twice as important parameter • Reduction of number of inputs seems not possible Part 4: Multidisciplinary Optimization

  4. Example II: Gradient-based optimization • Best design with valid constraints: mass = 1595 (19% of initial mass) • Areas of elements 2,5,6 and 10 are set to minimum • Stresses in remaining elements reach maximum value • 153 solver calls (+100 from DOE) Part 4: Multidisciplinary Optimization

  5. Example II: Adaptive response surface • Best design with valid constraints: mass = 1613 (19% of initial mass) • Areas of elements 2,6 and are set to minimum, 5 and 10 are close to minimum • 360 solver calls Part 4: Multidisciplinary Optimization

  6. Example II: EA (global search) • Best design with valid constraints: mass = 2087 (25% of initial mass) • 392 solver calls Part 4: Multidisciplinary Optimization

  7. Example II: EA (local search) • Best design with valid constraints: mass = 2049 (24% of initial mass) • 216 solver calls (+392 from global search) Part 4: Multidisciplinary Optimization

  8. Example II: Overview optimization results • NLPQL with small differentiation interval with best DOE as start design is most efficient • Local ARSM gives similar parameter set • EA/GA/PSO with default settings come close to global optimum • GA with adaptive mutation has minimum constraint violation Part 4: Multidisciplinary Optimization

  9. When to use which optimization algorithms • Gradient-based algorithms • Most efficient method if gradients are accurate enough • Consider its restrictions like local optima, only continuous variablesand noise • Response surface method • Attractive method for a small set of continuous variables (<15) • Adaptive RSM with default settings is the method of choice • Biologic Algorithms • GA/EA/PSO copy mechanisms of nature to improve individuals • Method of choice if gradient or ARSM fails • Very robust against numerical noise, non-linearities, number of variables,… Start Part 4: Multidisciplinary Optimization

  10. Sensitivity Analysis and Optimization 1) Start with a sensitivity study using the LHS Sampling • 2) Identify the important parameters and responses • understand the problem • reduce the problem Scan the whole Design Space Understand the Problem using CoP/MoP optiSLang Search for Optima 3) Run an ARSM, gradient based or biological based optimization algorithm 4) Goal: user-friendly procedure provides as much automatism as possible Part 4: Multidisciplinary Optimization

  11. Optimization of a Large Ship Vessel • Optimization of the total weight of two load cases with constrains (stresses) • 30.000 discrete Variables • Self regulating evolutionary strategy • Population of 4, uniform crossover for reproduction • Active search for dominant genes with different mutation rates Solver: ANSYS Design Evaluations: 3000 Design Improvement: > 10 % EVOLUTIONARY ALGORITHM Part 4: Multidisciplinary Optimization

  12. Optimization of passive safety Adaptive Response Surface Methodology • Optimization of passive safety performance US_NCAP & EURO_NCAP • using Adaptive Response Surface Method • 3 and 11 continuous variables • weighted objective function • Solver: MADYMO Design Evaluations: 75 Design Improvement: 10 % Part 4: Multidisciplinary Optimization

  13. Solver: ANSYS (using automatic spot weld Meshing procedure) Design evaluations: 200 Design improvement: 47% 134 binary variables, torsion loading, stress constrains Weak elitism to reach fast design improvement Fatigue related stress evaluation in all spot welds Genetic Optimization of Spot Welds Part 4: Multidisciplinary Optimization

  14. Optimization of an Oil Pan The intention is to optimize beads to increase the first eigenfrequency of an oil pan by more than 40%. Topology optimization indicate possibility > 40% improvement, but test failed. Sensitivity study and parametric optimization using parametric CAD design + ANSYS workbench+optiSLang could solve the task. Initial design beads design after topology optimization beads design after parameter optimization Design Parameter 50 Design Evaluations: 500 CAE: ANSYS workbench CAD: Pro/ENGINEER [Veiz. A; Will, J.: Parametric optimization of an oil pan; Proceedings Weimarer Optimierung- und Stochastiktage 5.0, 2008] Part 4: Multidisciplinary Optimization

  15. Several optimization criteria are formulated in terms of the input variables x Strategy A: Only the most important objective function is used as optimization goal Other objectives as constraints Strategy B: Weighting of single objectives Multi Criteria Optimization Strategies Part 4: Multidisciplinary Optimization

  16. Example: damped oscillator • Objective 1: minimize maximum amplitude after 5s • Objective 2: minimize eigen-frequency • DOE scan with 100 LHS samples gives good problem overview • Weighted objectives require about 1000 solver calls Part 4: Multidisciplinary Optimization

  17. Multi Criteria Optimization Strategies Strategy C: Pareto Optimization Part 4: Multidisciplinary Optimization

  18. Multi Criteria Optimization Strategies Design space Objective space • Only for conflicting objectives a Pareto frontier exists • For positively correlated objective functions only one optimum exists Part 4: Multidisciplinary Optimization

  19. Multi Criteria Optimization Strategies Conflicting objectives Correlated objectives Part 4: Multidisciplinary Optimization

  20. Multi Criteria Optimization Strategies Pareto dominance • Solution a dominates solution c since a is better in both objectives • Solution a is indifferent to b since each solution is better than the respective other in one objective (a dominates c) (a is indifferent to b) Part 4: Multidisciplinary Optimization

  21. Multi Criteria Optimization Strategies Pareto optimality • A solution is called Pareto-optimal if there is no decision vector that would improve one objective without causing a degradation in at least one other objective • A solution a is called Pareto-optimal in relation to a set of solutions A, if it is not dominated by any other solution c Requirements for ideal multi-objective optimization • Find a set of solutions close to the Pareto-optimal solutions (convergence) • Find solutions which are diverse enough to represent the whole Pareto front (diversity) Part 4: Multidisciplinary Optimization

  22. Multi Criteria Optimization Strategies Pareto Optimization using Evolutionary Algorithms • Only in case of conflicting objectives, a Pareto frontier exists and Pareto optimization is recommended (optiSLang post processing supports 2 or 3 conflicting objectives) • Effort to resolute Pareto frontier is higher than to optimize one weighted optimization function Part 4: Multidisciplinary Optimization

  23. Example: damped oscillator • Pareto optimization with EA gives good Pareto frontier with 123 solver calls Part 4: Multidisciplinary Optimization

  24. Example II: linear truss structure Anthill plot from ARSM Pareto front 1. • For more complex problems the performance of the Pareto optimization can be improved if a good start population is available • This can be found in selected designs of a previous DOE or single objective optimization Part 4: Multidisciplinary Optimization

  25. Optimization Algorithms Local adaptive RSM Biologic Algorithms Genetic algorithms, Evolutionary strategies & Particle Swarm Optimization Gradient-based algorithms Start Response surface method (RSM) Global adaptive RSM Pareto Optimization Part 4: Multidisciplinary Optimization

More Related