Optimization of Cure Cycles for Thermosetting Composites Fabrication under Uncertainty

1. 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

2. Outline • Introduction • Process Description • Process Uncertainties • Objectives • Process Model • Solution Approach • Quantification of Uncertainties • Optimization under Uncertainties • Parametric studies • Results • Model Validations • Parametric effects on the optimization results • Optimum temperature cycles

3. Process Description T Resin-saturated fiber Fiber z • 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. • An important process parameter is the cure temperature cycle (or, simply, cure cycle)–the magnitude and duration of temperature variation. • Determination of optimal cure cycle is critical to ensure product quality while simultaneously minimizing manufacturing time. • 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. Resin Bath Shape Preformer Puller Cut-Off Product Pultrusion Die Let-off Curing

4. 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 Key Sources of Process Uncertainties

5. Objectives • To develop a stochastic modeling framework for systematic inclusion of uncertainties in the cure process modeling • To determine the optimum cure cycles accounting for these uncertainties • To conduct systematic parametric studies to assess the influence of uncertainties and of constraints on process designs

6. Deterministic Model Sampler Output Variabilities Input Parameters Optimum Design Stochastic Model Optimizer Schematic of Optimization under Uncertainty The Main Steps: • The input parameter uncertainties are represented by a distribution type (Gaussian, Lognormal, etc.) and quantified in terms of mean and variance. • The deterministic process model serves as the basis of the approach, where the uncertainties propagates to shape the output distribution • An optimization scheme is carried out to obtain the optimum design. The deterministic model forms the core of the method and is discussed first

7. Process Model • 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.

8. Quantification of Uncertainties • Uncertain input parameters: (1) Cure Cycle – temperature magnitudes Ti (2) Kinetic Parameters –K1, E1, K2, E2 • These uncertainties are represented as Gaussian distributions with mean m, and standard deviation s. • Define: coefficient of variances/m • quantifies the severity of uncertainty in the parameters • the “width” of the distribution • Variable output parameters are: (1) Cure Time, tcure (2) Maximum Temperature, Tmax (3) Maximum Temperature Difference, DTmax (4) Minimum Degree of Cure, amin • Quantification of output variability • 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

9. The Optimization Problem Formulation • The optimization problem: Cure Time • 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. • Critical temperature Tcritcontrols the residual stress in the product which depends mostly on maximum temperature • Temperature difference constraint ensures temperature homogeneity which controls product’s uniformity of property. • The degree of cure a must be greater than a critical value in order to guarantee complete curing of the composite. Probabilistic Constraints

10. Decision Variables: The Cure Cycle The cure cycle is considered to be a 4-stage variation specified in terms of: (a) End-point temperatures: T1 , T2 , T3 , T4 (b) Stage time durations: t1 , t2 , t3 , t4 (which are kept deterministic)

11. Evaluation of Objective Function and Constraints SAMPLER Kinetic Parameters Latin Hypercube Sampling K1 , E1 K2 , E2 … KN , EN K,E Objective tcure Cure Time SAMPLER Constraints DETERMINISTIC PULTRUSION MODEL f(Tn,tn,Ki,Ei) T11…T1m T21…T2m … TN1…TNm Ti Tcrit Critical Temp. Tcrit Stage Duration Critical Temp. Diff. t1 … tm Temperature Cycle  Minimum Cure Optimization Parameters Confidence Level

12. Initial Guesses Stochastic Model (Figure on previous slide) Objective Function Simplex Search DecisionVariables Ti , ti ; i = 1,…,4 Stopping Criteria Reached? Metropolis Criterion No SIMULATED ANNEALING OPTIMIZER Yes Optimal Cure Cycles Optimization Schematic • Simulated Annealing (SA), a non-gradient based optimization, combined with simplex search algorithm is used to solve the optimization problem.

13. Optimization Algorithm • Simulated Annealing invokes the stochastic model for trial design determined through a simplex search • Acceptable design updates are determined through the Metropolis criterion • The sequence of design updates continues for a specified annealing temperature scheduleor until the design converges to a tolerance • The converged design is the optimum cure cycle

14. Parametric Studies • Two material systems considered: • Owens-Corning fiberglass-polyester system (OC-E701/P16N/BPO) and • American Cyanamid polyester system (CYCOM-4102) • The optimization was performed for different values of: • Critical temperature, Tmax • Critical temperature difference, DTmax • Pultrusion die diameter, D • Input coefficients of variance, s/m, and • Confidence level, pc

15. Pultrusion Model Validation • The pultrusion model is validated against the results in Han et. al. (1986) • The present simulation shows close agreement to the results in literature.

16. Determination of Number of Samples CYCOM-4102 • Number of samples for stochastic simulation is determined based on convergence of the standard deviation of the objective function and constraints. • Materials with faster resin kinetics require larger sample size • 50 samples for OC-E701/P16N/BPO • 300 samples for CYCOM-4102 300 Samples

17. Effects of Critical Temperature, Tcrit OC-E701/P16N/BPO CYCOM-4102 • Cure time increases as critical temperature decreases, and as coefficient of variance increases • For small critical temperatures, cure time increases significantly with the coefficient of variance • The increase is more pronounced for CYCOM-4102 which has faster kinetics.

18. Effects of Critical Temperature Difference, DTcrit OC-E701/P16N/BPO CYCOM-4102 • Cure time increases as maximum temperature difference decreases, and as coefficient of varianceincreases • For CYCOM-4102, at all levels of critical temperature difference, cure time increases significantly as coefficient of variance increases.

19. Effects of Confidence Level, pc,and Diameter, DOC-E701/P16N/BPO • Results shown for different diameter (D): • As s/mincreases (larger uncertainties), so doescure time • Cure timeincreases as diameterincreases, owing to the increased volume to be cured • For high desired confidence level, pc, the effect of coefficient of variation on cure time is more pronounced

20. Effects of Confidence Level, pc,and Diameter, DCYCOM-4102 • Effect of diameter (D) same as for the previous resin system: • Cure timeincreases as s/mincreases, and as diameter (D)increases. • The higher the confidence level, the greater the effect of coefficient of variation on the cure time. • The effect is more pronounced than in OC-E701/P16N/BPO, due to higher resin reactivity (faster kinetics) of CYCOM-4102

21. Optimum Cure Cycles: Effect of DTcrit CYCOM-4102 OC-E701/P16N/BPO • Plots show the effect of maximum temperature difference at specified stochastic parameters (s/m and pc). • The slope of the ramp in the cure cycles decreases as DTcritdecreases, thereby leading to longer cure time.

22. Optimum Cure Cycles: Effect of s/m CYCOM-4102 OC-E701/P16N/BPO • Plots show the effect ofcoefficient of variance on the cure cycles. • With increasingdegree of uncertainty (i.e. increasing s/m), the cure cycle timeincreases. The form of the cure cycles changes so that all process constraints are simultaneously satisfied.

23. Conclusions • An approach for optimization of cure cycles for thermosetting composites fabrication under uncertainty was developed. • Optimum cure cycles are presented for fabrication of two resin systems under uncertainty. • As critical temperature and critical temperature differenceincreases, cure cycle timedecreases, while as pultrusion die diameterincreases, so does the cure cycle time. • 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.