Sensitivity Analysis Multidisciplinary Optimization Robustness Evaluation and Robust

Sensitivity Analysis, Multidisciplinary Optimization, Robustness Evaluation, and Robust Design Optimization with opti. SLang 3. 2

Outline • Introduction • Process Integration • Parametrize editor • Interfaces to common solvers • Post processing • Sensitivity analysis • Design of experiments • Coefficient of correlation • Simple regression, quadratic & rank order correlation • Multiple regression, Coefficient of Determination (Co. D) • Coefficient of Importance (Co. I) • Significance filter • Moving Least Squares approximation • Coefficient of Prognosis (Co. P) • Meta-model of Optimal Prognosis (MOP) • Applications Ø Accompanying example: Sensitivity analysis of an analytical function (Tutorial 1) 2 Outline & Flowcharts

Outline • Multidisciplinary Optimization • Single objective, constraint optimization • Gradient based optimization • Global and adaptive response surface methods • Evolutionary algorithm (EA) • Particle swarm optimization (PSO) • Multi objective optimization • Pareto optimization with evolutionary algorithm • Applications Ø Accompanying example: Optimization of a damped oscillator (Tutorial 2, Part 1) • Model calibration/identification • Parametrization of characteristic curves as signals • Sensitivity analysis • Definition of objective functions • Dependent parameters Ø Accompanying example: Calibration of a damped oscillator (Tutorial 2, Part 2) 3 Outline & Flowcharts

Outline • Robustness analysis • Definition of robustness • Random variables • Definition of uncertainties • Variance-based robustness analysis • Statistical measures • Applications • Reliability analysis Ø Accompanying example: Robust design optimization of a damped oscillator (Tutorial 2, Part 3) • Robust design optimization • Definition of robust design optimization (RDO) • Design for Six-Sigma • Iterative RDO procedure • Applications • Simultaneous RDO procedure Ø Accompanying example: Robust design optimization of a damped oscillator (Tutorial 2, Part 3) 4 Outline & Flowcharts

Standard optimization Optimization Sensitivity analysis DOE Solver • • • 5 MOP Optimizer • Gradient • ARSM • EA/GA Solver Full design variable space X for sensitivity analysis Scanning the design space with DOE by direct solver calls Generating MOP on DOE samples Sensitivity analysis gives reduced design variable space Xred Optimization requires start value x 0, objective function f(x) and constraint conditions gj(x) Optimizer determines optimal design xopt by direct solver calls Outline & Flowcharts

Optimization with MOP pre-search Optimization Sensitivity analysis DOE MOP Solver Optimizer • Gradient • ARSM • EA/GA MOP Solver • Full optimization is performed on MOP by approximating the solver response • Optimal design on MOP can be used as • final design (verification with solver is required!) • as start value for second optimization step with direct solver • Good approximation quality of MOP is necessary for objective and constraints (Co. P ≥ 90%) 6 Outline & Flowcharts

Optimization with MOP using external DOE Optimization External DOE Sensitivity analysis Excel plugin MOP Optimizer • Gradient • ARSM • EA/GA MOP • External DOE exists from experiments or other sources • Excel plugin is used to generate opti. SLang binary file • MOP uses external DOE scheme to generate approximation and to perform sensitivity analysis • Optimization is performed on MOP to obtain approximate optimum 7 Outline & Flowcharts

Optimization + Robustness evaluation Sensitivity analysis DOE MOP Solver • • 8 Optimization Robustness Optimizer Robustness • Variance • Sigma-level • Reliability • Gradient • ARSM • EA/GA Solver Full optimization variable space X for sensitivity analysis Sensitivity analysis gives reduced optimization variable space Xred Optimizer determines optimal design xopt by direct solver calls Robustness evaluation (varianced-based or reliability-based) in the random variable space Xrob at optimal design xopt Outline & Flowcharts

Iterative Robust Design Optimization Sensitivity analysis DOE MOP Solver Optimization Robustness Optimizer Robustness • Variance • Sigma-level • Reliability • Gradient • ARSM • EA/GA Solver No Update constraints Yes End • Sensitivity analysis gives reduced optimization variable space Xred • Optimizer determines optimal design xopt by direct solver calls • Robustness evaluation • Robust optimum – end of iteration • Non-robust optimum - update constraints and repeat optimization + robustness evaluation 9 Outline & Flowcharts

Simultaneous Robust Design Optimization Sensitivity analysis DOE Optimizer MOP Solver Robustness Solver • Sensitivity analysis gives reduced optimization variable space Xred • Optimizer determines optimal design xopt by direct solver calls with simultaneous robustness evaluation for every design • Each robustness evaluation determines robustness values by direct solver calls 10 Outline & Flowcharts