Optimization of Cure Cycles for Thermosetting Composites Fabrication
Optimization of Cure Cycles for Thermosetting Composites Fabrication under Uncertainty A. Mawardi and R. Pitchumani Composites Processing Laboratory Department of Mechanical Engineering University of Connecticut Storrs, Connecticut http: //www. engr. uconn. edu/cml Sponsors: Office of Naval Research, National Science Foundation Presented at the INFORMS 2001 Conference, Maui Hawaii
Outline q Introduction Ø Process Description Ø Process Uncertainties Ø Objectives q Process Model q Solution Approach Ø Quantification of Uncertainties Ø Optimization under Uncertainties Ø Parametric studies q Results Ø Model Validations Ø Parametric effects on the optimization results Ø Optimum temperature cycles Composites Processing Laboratory, University of Connecticut
Process Description Fiber Let-off Resin-saturated fiber Resin Bath Shape Preformer T z Pultrusion Die Puller Cut-Off Product Curing q Process considered is the fabrication of thermosetting-matrix composites by pultrusion. The primary processing step is that of curing of a fiber-resin mixture within a heated die to produce the composites. q An important process parameter is the cure temperature cycle (or, simply, cure cycle)–the magnitude and duration of temperature variation. q Determination of optimal cure cycle is critical to ensure product quality while simultaneously minimizing manufacturing time. q Optimal cure cycle design is often obtained using deterministic models. However, process uncertainties exist in practice, which must be accounted for in the cure cycle design, which is the focus of the study. Composites Processing Laboratory, University of Connecticut
Key Sources of Process Uncertainties Materials: Uncertainties in characterizing material properties, such as kinetic parameters, … Design: Inaccuracy associated with description of process phenomena, process model, etc. Variable Product Quality Operational: Uncertainties in parameter setting, monitoring and control Composites Processing Laboratory, University of Connecticut
Objectives q To develop a stochastic modeling framework for systematic inclusion of uncertainties in the cure process modeling q To determine the optimum cure cycles accounting for these uncertainties q To conduct systematic parametric studies to assess the influence of uncertainties and of constraints on process designs Composites Processing Laboratory, University of Connecticut
Schematic of Optimization under Uncertainty Sampler Input Parameters Deterministic Model Stochastic Model Output Variabilities Optimum Design Optimizer The Main Steps: q The input parameter uncertainties are represented by a distribution type (Gaussian, Lognormal, etc. ) and quantified in terms of mean and variance. q The deterministic process model serves as the basis of the approach, where the uncertainties propagates to shape the output distribution q An optimization scheme is carried out to obtain the optimum design. The deterministic model forms the core of the method and is discussed first Composites Processing Laboratory, University of Connecticut
Process Model q Two dominant physical phenomena take place within the pultrusion die: (1) Heat transfer associated with heating from the die wall (2) The chemical reaction leading the cure process which are represented by the following equations: (1) (2) with boundary conditions: The coupled system of equations are solved using a finite difference scheme. Composites Processing Laboratory, University of Connecticut
Quantification of Uncertainties q Uncertain input parameters: (1) Cure Cycle – temperature magnitudes Ti (2) Kinetic Parameters – K 1, E 1, K 2, E 2 q These uncertainties are represented as Gaussian distributions with mean m, and standard deviation s. Ø Define: coefficient of variance s/m – quantifies the severity of uncertainty in the parameters – the “width” of the distribution q Variable output parameters are: (1) Cure Time, tcure (2) Maximum Temperature, Tmax (3) Maximum Temperature Difference, Tmax (4) Minimum Degree of Cure, min q Quantification of output variability Ø Define: confidence level (pc) as the percentage of samples for which the output parameters (i. e. cure time) lies below a specified value. Example: pc = 50% median pc = 100% extreme Composites Processing Laboratory, University of Connecticut
The Optimization Problem Formulation q The optimization problem: Cure Time Probabilistic Constraints q Tn(tn) represent end point temperatures of an n-stage piecewise linear cure cycle, and tcure is the time needed for the composite to cure. q Critical temperature Tcrit controls the residual stress in the product which depends mostly on maximum temperature q Temperature difference constraint ensures temperature homogeneity which controls product’s uniformity of property. q The degree of cure a must be greater than a critical value in order to guarantee complete curing of the composite. Composites Processing Laboratory, University of Connecticut
Decision Variables: The Cure Cycle The cure cycle is considered to be a 4 -stage variation specified in terms of: (a) End-point temperatures: T 1 , T 2 , T 3 , T 4 (b) Stage time durations: t 1 , t 2 , t 3 , t 4 (which are kept deterministic) Composites Processing Laboratory, University of Connecticut
Evaluation of Objective Function and Constraints SAMPLER Latin Hypercube Sampling K, E K 1 , E 1 K 2 , E 2 … KN , E N Kinetic Parameters Objective tcure SAMPLER T 11…T 1 m T 21…T 2 m … TN 1…TNm Ti Stage Duration DETERMINISTIC PULTRUSION MODEL f(Tn, tn, Ki, Ei) t 1 … t m Temperature Cycle Constraints Tcrit Optimization Parameters Cure Time Critical Temp. Diff. Minimum Cure Confidence Level Composites Processing Laboratory, University of Connecticut
Optimization Schematic q Simulated Annealing (SA), a non-gradient based optimization, combined with simplex search algorithm is used to solve the optimization problem. Initial Guesses Stochastic Model (Figure on previous slide) Objective Function Simplex Search Decision Variables Ti , ti ; i = 1, …, 4 Stopping Criteria Reached? Metropolis Criterion No SIMULATED ANNEALING OPTIMIZER Yes Optimal Cure Cycles Composites Processing Laboratory, University of Connecticut
Optimization Algorithm q Simulated Annealing invokes the stochastic model for trial design determined through a simplex search q Acceptable design updates are determined through the Metropolis criterion q The sequence of design updates continues for a specified annealing temperature schedule or until the design converges to a tolerance q The converged design is the optimum cure cycle Composites Processing Laboratory, University of Connecticut
Parametric Studies q Two material systems considered: ü Owens-Corning fiberglass-polyester system (OC-E 701/P 16 N/BPO) and ü American Cyanamid polyester system (CYCOM-4102) q The optimization was performed for different values of: Ø Critical temperature, Tmax Ø Critical temperature difference, Tmax Ø Pultrusion die diameter, D Ø Input coefficients of variance, s/m, and Ø Confidence level, pc Composites Processing Laboratory, University of Connecticut
Pultrusion Model Validation q The pultrusion model is validated against the results in Han et. al. (1986) q The present simulation shows close agreement to the results in literature. Composites Processing Laboratory, University of Connecticut
Determination of Number of Samples CYCOM-4102 q Number of samples for stochastic simulation is determined based on convergence of the standard deviation of the objective function and constraints. q Materials with faster resin kinetics 300 Samples require larger sample size Ø 50 samples for OC-E 701/P 16 N/BPO Ø 300 samples for CYCOM-4102 Composites Processing Laboratory, University of Connecticut
Effects of Critical Temperature, Tcrit OC-E 701/P 16 N/BPO CYCOM-4102 q Cure time increases as critical temperature decreases, and as coefficient of variance increases q For small critical temperatures, cure time increases significantly with the coefficient of variance q The increase is more pronounced for CYCOM-4102 which has faster kinetics. Composites Processing Laboratory, University of Connecticut
Effects of Critical Temperature Difference, DTcrit OC-E 701/P 16 N/BPO CYCOM-4102 q Cure time increases as maximum temperature difference decreases, and as coefficient of variance increases q For CYCOM-4102, at all levels of critical temperature difference, cure time increases significantly as coefficient of variance increases. Composites Processing Laboratory, University of Connecticut
Effects of Confidence Level, pc, and Diameter, D OC-E 701/P 16 N/BPO q Results shown for different diameter (D): Ø As s/m increases (larger uncertainties), so does cure time Ø Cure time increases as diameter increases, owing to the increased volume to be cured q For high desired confidence level, pc, the effect of coefficient of variation on cure time is more pronounced Composites Processing Laboratory, University of Connecticut
Effects of Confidence Level, pc, and Diameter, D CYCOM-4102 q Effect of diameter (D) same as for the previous resin system: Ø Cure time increases as s/m increases, and as diameter (D) increases. q The higher the confidence level, the greater the effect of coefficient of variation on the cure time. q The effect is more pronounced than in OC-E 701/P 16 N/BPO, due to higher resin reactivity (faster kinetics) of CYCOM-4102 Composites Processing Laboratory, University of Connecticut
Optimum Cure Cycles: Effect of DTcrit OC-E 701/P 16 N/BPO CYCOM-4102 q Plots show the effect of maximum temperature difference at specified stochastic parameters (s/m and pc). q The slope of the ramp in the cure cycles decreases as DTcrit decreases, thereby leading to longer cure time. Composites Processing Laboratory, University of Connecticut
Optimum Cure Cycles: Effect of s/m OC-E 701/P 16 N/BPO CYCOM-4102 q Plots show the effect of coefficient of variance on the cure cycles. q With increasing degree of uncertainty (i. e. increasing s/m), the cure cycle time increases. The form of the cure cycles changes so that all process constraints are simultaneously satisfied. Composites Processing Laboratory, University of Connecticut
Conclusions q An approach for optimization of cure cycles for thermosetting composites fabrication under uncertainty was developed. q Optimum cure cycles are presented for fabrication of two resin systems under uncertainty. q As critical temperature and critical temperature difference increases, cure cycle time decreases, while as pultrusion die diameter increases, so does the cure cycle time. q An increase in coefficient of variance (increasing degree of uncertainty) causes an increase in cure cycle time. This effect is more pronounced at more tightly constrained process, and at higher confidence level. Composites Processing Laboratory, University of Connecticut
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