A method for finding the optimal autoclave temperature history for reducing residualstresses in manufactured pre-impregnated thermosetting composites is presented. A cou-pled finite element model which incorporates a thermo-chemical and incremental elasticanalysis is used to predict the residual stress distribution at the edge of a thick compositebeam. The optimal autoclave temperature is sought by formulating an objective functionand an associated set of constraints. The objective is designed to maximize the failure loadof the manufactured beam subjected to an axial force, while the constraints are imposedto ensure that the composite is uniformly cured and does not sustain temperature damageduring the manufacturing process. A semi-analytic technique is presented for calculatingthe objective and constraint gradients. SNOPT, a powerful optimization program, is usedto find the optimal temperature history. Typical results are presented and future lines ofinvestigation are discussed.
I. Introduction
Thermal residual stresses have long been known to significantly affect the failure behaviour of compositestructures.1 Usually the largest residual stresses occur in a thin layer close to the edge of a composite
as these stresses adjust to meet the boundary conditions at the free surface. It is the stress concentrationsin this region which often cause failure. Thus, modifying the residual stress distribution by changing themanufacturing process conditions is one possible way to improve the performance of composite components.
Unger and Hansen2 and Unger3 improved the failure performance of thermoplastic beams by applyinga local reconsolidation to the edge of a beam after the completed manufacturing process. The reprocessingstep modified the residual stress distribution often causing a change in both the failure load and the modeof failure. Domb4 performed an analysis of the modified manufacturing process to predict the residual stressdistribution.
Modifying the stress distribution in thermosetting composites using a local reconsolidation technique isnot possible as the cure reaction is irreversible. However this technique is the motivation for the current workas it should be possible to obtain a more favourable residual stress distribution by modifying the processingcycle in an analogous fashion to the reconsolidation technique.
Several authours have presented different methodologies for optimizing the manufacturing process for pre-impregnated thermosetting composites.5–9 Typically, the time dependent autoclave processing conditionssuch as temperature and or pressure are sought. However, there is no general consensus on the formthat the objective should take. This reflects the many different competing design considerations whichneed to be taken into account when designing the manufacturing process for composite components. Suchconsiderations include, manufacturing time and cost, performance of the final component and reproducibilityof the composite part. Li et al.5 and Li and Tucker,6 sought to minimize the time required to manufacturethe composite, while Gopal et al.8 sought to minimize the residual stress moments in composite components.
The approach taken here is to form an objective based on a detailed residual stress distribution at theedge of the cured composite in order to improve the failure characteristics of the component. Although there
∗Graduate Student, Institute for Aerospace Studies, 4925 Dufferin St., AIAA Student Member†Professor, Institute for Aerospace Studies, 4925 Dufferin St., AIAA Member
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11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference6 - 8 September 2006, Portsmouth, Virginia
are many different manufacturing configurations, the analysis here is restricted to a long, rectangular beamloaded in the axial direction. As a result of the problem symmetry, it is only necessary to model half thecross section. This is a non-trivial representative problem which will reflect much of the physics of morecomplicated manufacturing setups. The remaining sections present some details of the analysis, a descriptionof the optimization problem and some results for various configurations.
Tooling
Composite
Bleeder Cloth Vacuum Bag
Bleeder Dam
Figure 1. Schematic of of the composite manufacturing layup.
II. The Process Model
Manufacturing thermosetting preimpregnated composite components involves many simultaneous coupledphysical processes. These processes include heat transfer from the autoclave to the interior of the composite, acomplex heat activated exothermic polymeric reaction, resin flow into the surrounding layup, the rheology ofthe epoxy, the development of the composite modulus as the epoxy hardens and the development of residualstrains in the matrix and fibers due to heat transfer and chemical processes.10 Each of these physicalprocesses have been modelled by different authors. Loos and Springer10 predicted the degree of cure in thecomposite as well as the resin flow into the surrounding layup and compaction of the ply layers based on arheological analysis of the resin. Bogetti and Gillespie12 modelled the residual stresses based on an analysisof the degree of cure and a prediction of the constitutive equations based on that calculation. Johnston etal.11 following the work of Bogetti and Gillespie, predicted the spring back in composite components.
The process model employed here follows from the development of Bogetti and Gillespie12 and Johnston etal.11 The model produces a detailed through thickness residual stress distribution. The main processes whichare included are a coupled thermo-chemical analysis and the residual stress development. The interactionbetween these phenomenon is shown schematically in Figure 2.
ThermalAnalysis
Residual StressAnalysis
Boundary Temperature
Post Processing
Thermal State
DisplacementState
Figure 2. Diagram of the complete manufacturing analysis
A. The Thermo-chemical Model
Two physical processes are included in the thermo-chemical model: heat transfer and a complex exothermicchemical reaction. The analysis of the chemical process is handled using a phenomenological model whichpredicts the amount of heat evolved from a given point in the epoxy based on the temperature and thedegree of cure. The phenomenological model captures the macroscopic properties of the reaction withoutdealing with the complex chemical reactions that take place in the epoxy. It is necessary to couple thechemical and thermal processes as the reaction is strongly exothermic and heat activated. There are manydifferent phenomenological models that have been developed for modelling the epoxy reaction, but most takean Arhenius form.10,12
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The cure kinetics model and heat transfer equations form a system of non-linear partial differentialequations in space and time. A straightforward Galkerin finite element formulation is used to discretize thespacial operators using bi-quadratic shape functions. These equations have been developed before by variousauthours and so the details are omitted here.5,6, 13,14 The discretization results in a non-linear system ofODEs. The trapezoidal time stepping method is employed to solve these equations time accurately whichresults in the following equation which must be solved implicitly at each time step:
Rk(xk,xk−1) = C[xk − xk−1
∆t
]+
12
[R0(xk) + R0(xk−1)
]= 0 (1)
where xk is the vector of thermo-chemical state variables at time step k, R0(xk) are the residuals of thefinite element formulation and C is a coupling matrix.
-0.02 0 0.02 0.04 0.06 0.08Y
-0.03
-0.02
-0.01
0
0.01
0.02
Z
Figure 3. Finite element discretization for the thermo-chemical analysis, displaying the different parts of thelayup shown in Figure 1.
A Newton-Krylov approach is used to solveEquation 1 at each time step.15 The generalizedminimal residual (GMRES) iterative method16 isused to approximately solve the resulting system oflinear equations with an incomplete LU precondi-tioner with level of fill k (ILU(k)). The ILU(k) fac-torization is frozen after the first Newton iteration.In regions where the residuals of the cure equationsexceed some specified tolerance a dual time march-ing technique is engaged. This improves the robust-ness of the solver but slows down the rate of conver-gence. The overall computational cost is not affectedsignificantly as this technique is only used for a fewtime steps during a complete cure simulation.
A representative finite element mesh used in thethermo-chemical calculations is presented in Figure3. The surrounding layup material conducts heat poorly and thus plays an important role in the thermo-chemical problem.
B. The Residual Stress Model
The residual stress model is dependent on the thermo-chemical analysis through the constitutive relations.As the cure cycle progresses, the resin hardens and the modulus of the composite develops. If viscoelasticeffects are neglected, the distribution of residual stresses can be determined solely from the interactionbetween the development of residual strains and material modulus.
The instantaneous constitutive properties of the resin-fiber system are determined using the Halpin-Tsaimicromechanics model17 which requires the mechanical properties of the resin and fiber separately. Themechanical properties of the resin are modelled in terms of the degree of cure and temperature while themechanical properties of the fibers are assumed to be constant during the entire cure process. In a similarfashion, the macroscopic residual strains are determined from the cure shrinkage and thermal expansion inthe fibers and resin.
A finite element model is used to calculate the incremental stresses over each time step. A uniform axialextension model is used to compute the invariant state of stress in the axial direction.11,12,18–20 This analysisassumes that all stresses and strains are invariant in the longitudinal direction. The total residual stressesare calculated as the sum of the incremental stresses developed over all time steps. A stretched mesh andbi-cubic elements are used to accurately capture the rapidly varying stresses in the vicinity of the free edge.
For ease of presentation the incremental residual stress problem is presented in the following form:
where ∆uk are the incremental displacements over the kth time step, Sk is the global stiffness matrix andFk is the global force vector due to the residual strains.
After the final time step, an additional calculation is performed to determine the stress state in the beamdue to a unit axial load. The combination of the residual stresses and the stresses induced by the unit axialload are used to predict the failure load of the entire beam.
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III. The Optimization Problem
The optimization problem is defined in this section by specifying the problem design variables andformulating the objective function and constraints.
The design variables define a set of control points which are used to discretize the autoclave boundarytemperature in time. The boundary temperature is interpolated between these points using either a cubicspline or a piecewise linear interpolation. Other types of discretization are possible, such as a series oftemperature dwells where the dwell temperature itself would be permitted to vary. It should be emphasizedthat the discretization of the boundary temperature automatically imposes an implicit set of constraints onthe optimization problem by restricting the temperature distribution to the function space defined by thediscretization.
0 50 100 150Time [min]
300
350
400
450
500
Tem
pera
ture
[K]
Boundary TemperatureControl Points
(a) Piecewise Linear Interpolation
0 50 100 150Time [min]
300
350
400
450
500
Tem
pera
ture
[K]
Boundary TemperatureControl Points
(b) Spline Interpolation
Figure 4. Two possible interpolations between control points.
The objective function is designed to favour an increase in the beam’s axial failure load during actual lifecycle usage. The stress state in the beam is therefore assumed to be a combination of the residual stressesinduced during the processing stage and the stresses induced by an applied axial load. The failure criterionis applied throughout the domain of the composite by first applying it to each element individually. Theglobal failure load is then determined by taking the minimum of this set of values. The potential problemwith this type of calculation is that the resulting failure load is non-differentiable and may yield poor resultswhen combined with a gradient based optimization algorithm. In order to circumvent this problem, theKreisselmeier-Steinhauser (KS) aggregation technique has been employed,21,22 which smoothes the non-differentiability of this type of max-min optimization problem. The objective is then defined as a functionof an integral over the domain of the composite in the following manner:
I(y) = −Gmin − 1ρ
ln[∫
Ω
exp−ρ
[G(σR, σA)−Gmin
]dΩ
](3)
where y is the vector of design variables, ρ is a user defined parameter typically taken as 30 and σRi and σA
i
are the residual and axial stresses respectively, both expressed in the the material axis. G(σR, σA) can be anarbitrary function but is intended to approximate the pointwise failure load. Lastly, Gmin is the minimumvalue of G over the domain of the composite.
There are several possible formulations for the function G(σR, σA). In the following work this pointwisefunction takes the form:
G(σR, σA) =Xm − σR
m
σAm
(4)
where m is the maximum stress failure mode that is active for the original design and Xm is the positive ornegative failure stress depending on the sign of σA
m. This function is only non-differentiable when σAm = 0.
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In all cases presented here, the original and optimized beams fail in the same mode. However, there could bea situation in which two failure modes are active at the optimal solution. Equation 4 would not be capableof determining the failure load in such a situation and it would be necessary to use the full maximum stressfailure criterion:
G(σR, σA) = min
Xi − σRi
σAi
i = 1, . . . 6 (5)
The problem with Equation 5 is that it is non-differentiable at points where two failure loads are equal which,in this hypothetical situation, is the location of the optimal design. This problem could be circumvented byusing a different failure criterion or by smoothing the non-differentiable points of this function perhaps byusing a KS function.
The end result of these definitions is that at the optimal solution, as ρ →∞, the minimum failure load,as calculated by the function G(σR, σA), is maximized. For large but finite values of ρ a max-min problemformulation is approached.21 It should be noted that more loading configurations could be included in thedesign such as stresses induced by a compression, bending or shearing load. These stress states would furthercomplicate the optimization problem as a single failure load would no longer characterize the performanceof the beam.
Following the discussion of Loos and Springer10 two constraints are imposed on the composite manufac-turing process: first, that the maximum temperature in the interior of the composite must not exceed somepreset temperature during the entire cure cycle; second, that the composite must be cured uniformly andcompletely. The temperature constraint can be written in the following form:
max T (y, z, t) ≤ Tmax (6)
where Tmax is a preset maximum temperature, and T (y, z, t) is the temperature distribution over all time.The cure constraint can be written in the following form:
min c(y, z, tfinal) ≥ cmin (7)
where cmin is the minimum required degree of cure and tfinal is the time at the completion of the cure cycle.Both of these constraints are imposed on the thermo-chemical problem directly and do not influence theresidual stress problem. In both instances a KS constraint aggregation technique is used to approximatelyenforce the constraints.21,23
IV. Gradient Evaluation
The following section outlines the development of the mixed direct-adjoint formulation for the semi-analytic computation of the gradient of the objective function. Conveniently, the sensitivities of the thermalstate variables are generated through this method and so the gradients of the constraints can be calculatedin tandem with the objective gradient evaluation.
The complete method can be described, in abstract terms, in the following manner. The objectivefunction is dependent on the residual stresses which are controlled through the thermo-chemical process.This process in turn, is controlled through the boundary temperature in the autoclave. The sensitivities ofthe thermal state variables are computed using the direct method at each time step. This determines thevariation of the thermal state with respect to a change in the boundary temperature. Next, the residualstress problem is treated as if each of the thermal state variables is itself a control variable. An adjointformulation is used to determine the contribution of the thermal residual stress problem to the sensitivity ofthe objective.
The first step in the development of this computation is to express the sensitivity of the objective functionwith respect to the design variables. It should be stressed here that the objective function is dependent onthe thermal and residual stress state variables at all time steps. This yields the following equation:
dI
dy=
∂I
∂y+
n∑
k=1
[∂I
∂xk
∂xk
∂y+
∂I
∂∆uk
∂∆uk
∂y
](8)
The difficulty in the computation of the gradient lies in the terms ∂xk/∂y and ∂∆uk/∂y which are thesensitivities of the state variables to the design variables. Both of these terms are handled in different ways.
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In order to deal with the thermal state variation, it is necessary to consider the residuals of the time steppingscheme at each iteration. Since the residual is zero upon convergence to the solution, the sensitivity of theresiduals with respect to the design variables must be zero. This can be expressed in the following manner:
dRk
dy=
∂Rk
∂y+
∂Rk
∂xk
∂xk
∂y+
∂Rk
∂xk−1
∂xk−1
∂y= 0 (9)
The direct sensitivities of the thermal state variables are determined by solving Equation 9, which yields thefollowing result for the trapezoidal time marching technique:
∂xk
∂y= −
[C∆t
+12
∂R0
∂xk
]−1 ∂R0
∂y+
[− C
∆t+
12
∂R0
∂xk−1
]∂xk−1
∂y
(10)
It should be noted that this computation involves the solution of a large sparse matrix with a number ofdifferent right-hand sides: one for each design variable. For moderately sized thermal problems a direct LUfactorization was found to be highly effective since the cost of the factorization is amortized over each of thesystem solves.
The sensitivities of the residual stress problem are dealt with using an adjoint formulation. In a similarmanner to the thermo-chemical problem above, the total derivative of the residuals of the residual stressproblem is taken. This yields the following set of equations:
dPk
dy=
∂Pk
∂y+
∂Pk
∂xk
∂xk
∂y+
∂Pk
∂xk−1
∂xk−1
∂y+
∂Pk
∂∆uk
∂∆uk
∂y= 0 (11)
It is important to emphasize that the residual stress problem is not directly dependent on the boundarytemperature of the thermal problem. Thus, the gradient of the residual stress problem with respect to theboundary temperature on the thermo-chemical problem is identically zero, ∂Pk/∂y = 0. Applying thissimplification and solving for ∂∆uk/∂y, allows Equation 8 to be expressed as:
dI
dy=
∂I
∂y+
n∑
k=1
[∂I
∂xk
∂xk
∂y+ λT
k
∂Pk
∂xk
∂xk
∂y+
∂Pk
∂xk−1
∂xk−1
∂y
](12)
Where the derivatives ∂xk/∂y are computed from Equation 10 and where the adjoint variables λk are definedin the following manner:
λTk
∂Pk
∂∆uk= − ∂I
∂∆uk(13)
It should be noted that the adjoint formulation of the residual stress problem is only possible because theincremental displacement states at previous time steps do not affect the current incremental displacementstate. If a viscoelastic formulation were used, this would not be the case and the gradient would have to becomputed using an alternate method such as an adjoint involving all displacement and thermal states at alltime steps simultaneously.
A. Some Implementation Details
Following the work of Alonso et. al.,24 the objective and gradient evaluation routines have been interfaced toPython. In turn several different optimizers have been interfaced to Python, notably SNOPT,25 KNITRO26
and MMA.27 Typically, especially for legacy codes, this greatly simplifies the interface between variousoptimization algorithms and the analysis by neatly defining the interface between them. Python then actsas the glue language between high level function calls. There is almost no performance penalty with thistechnique as the analysis and gradient evaluation can still be performed in a low level programming language.
V. Results
The following section presents optimization results for a series of 8 laminated beams which are presentedin Table 1. The material characterization data is taken from Bogetti and Gillespie12 for a graphite/epoxysystem. The failure characterization data is taken from Jones.1 These are representative of typical graphitecomposite material properties.
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Each beam is subjected to a 2.5 hour cure cycle with a post-cure cooling period. A total of 110 timesteps are used to simulate the time dependent process. For all cases identical objective parameters havebeen used with ρ = 30 and m = 2 corresponding to a transverse matrix cracking failure mode. Furthermore,the temperature is constrained to be less than Tmax = 470K and the minimum degree of cure is constrainedto be greater than cmin = 0.95. For each case a cubic spline interpolation of the boundary temperaturedistribution is sought using 11 control points. The first and last control points are fixed while the remainingpoints are free to vary in temperature but are fixed in time. The convergence criteria used in SNOPT is thatthe infeasibility and optimality error must both be less than 10−6. A typical convergence plot from SNOPTis presented in Figure 5.
The optimization results from each of the 8 cases are summarized below. Table 2 summarizes the originaland optimal tensile failure loads and the relative improvement between the two. The relative improvementin the tensile failure load strongly depends on the beam construction with certain combinations of ply angleshaving inherently lower tensile failure loads. Table 3 compares the original and optimal compressive failureloads. These results suggest that the optimization process has either a negligible or favourable impact on thecompressive failure loads, even though these loads do not enter the optimization problem directly. Figures 6and 7 show each of the optimal boundary temperature distributions. It is interesting to note the similaritybetween the different optimal cure cycles. This similarity is advantageous as it would suggest that theoptimal cure cycle is relatively insensitive to the exact beam construction.
Figure 8 shows the complete σyz stress distribution before and after optimization for problem P3, demon-strating that the peak stresses are reduced while Figure 9 shows a slice of certain stress components in they direction before and after the optimization.
0 5 10 15 20Major Iterations
-2.36
-2.35
-2.34
-2.33
-2.32
-2.31
-2.3
Ob
ject
ive
10-8
10-6
10-4
10-2
100
Op
timal
ityE
rror
,In
feas
ibili
ty
ObjectiveOptimality ErrorInfeasibility
Figure 5. Convergence history for design problem P1
VI. Conclusions
A method was presented for determining the optimal time dependent autoclave processing conditionsbased on the final failure performance of the composite. The results from the simple test cases presentedhere suggest that the optimal temperature history is, to a certain extent, independent of the specific beamconstruction, but will have a more significant impact on the failure load depending on the order and orien-tations of the plies.
It should be noted that the results from this paper suggest that the improved performance of the com-
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pleted composite is often modest but that this improvement could be made at almost no cost. The resultingboundary temperature distributions do not require rapid heating or cooling rates, thus in many cases nonew infrastructure would be required.
One aspect of the manufacturing process not addressed in this paper is the duration of the cure cycleitself. It could be advantageous to vary the length of the cure to determine the tradeoffs between a fastercure cycle and the axial failure load. It would be possible to include the total cure time as a design variable,but it would almost certainly be necessary to impose a constraint or some sort of penalization to preventexcessively long cure cycles.
The change in tensile failure load is measured as a percentage of the original, unoptimized,failure load.
Acknowledgments
This research was supported by the Natural Sciences and Engineering Research Council of Canada andthe Ontario Graduate Scholarship program. The first authour gratefully acknowledges Professor J.R.R.A.Martins for his assistance advice and the use of the pyMDO optimization utilities.
References
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The change in compressive failure load is measured as a percentage of the original, unopti-mized, failure load.
0 50 100 150Time [min]
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Tem
pera
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[K]
Boundary TemperatureControl Points
(a) P1
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(d) P4
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(a) P5
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(d) P8
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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Y
-0.02
-0.01
0
0.01
0.02
Z
σyz
5.8314.1672.5030.839
-0.826-2.490-4.154-5.818
(a) Original
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Y
-0.02
-0.01
0
0.01
0.02
Z
σyz
5.8314.1672.5030.839
-0.826-2.490-4.154-5.818
(b) Optimized
Figure 8. Original and optimal σyz stress distribution results for problem P3.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07Y
-20
-15
-10
-5
0
5
10
15
20
25
Str
ess
[MP
a]
σxx
σyy
σxy
σxx
σyy
σxy
OptimalOriginal
Figure 9. A slice of the stress for problem P3.
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