Coupled Compression RTM andComposite Layup Optimization

R. Le Riche1, A. Saouab2 and J. Breard2

1 CNRS URA 1884 / SMS, Ecole des Mines de Saint Etienne,France, leriche@emse.fr (corresponding author).2 Lab. de Mecanique, Univ. du Havre, France.

With additional erratum on (SO) formulation as compared to thepublished version in Composites Science and Technology, Vol.63, No.15,

Elsevier publ., July 2003, pp. 2277-2287.

Abstract

A methodology for studying the implications of taking into accountmanufacturing at early design stages is presented. It is applied to arectangular laminate made by RTM and compression (RTCM). Theplate is designed for injection time, maximum mold pressure, stiff-ness and buckling. Semi-analytical, numerically inexpensive modelsof the processes and structure enable a thorough investigation of thecouplings between process and structure by comparing four design for-mulations: in decoupled problems, either the process or the structureis optimized; then, the process is optimized with targetted low or highstructural performance. A globalized Nelder-Mead optimization algo-rithm is used. The rigour of the method and the relative simplicity ofthe application case provide a clear description of how maximum moldpressures, injection times and final structures properties are traded-off.

Keywords : Optimization, Resin transfer moulding (E), Laminates (C),Porosity (B), Buckling (C).

1 Introduction

After having been praised for their mechanical properties, composite mater-ials have lately received much attention with regard to reducing their manu-facturing cost. The possibility of automating composite structure productionby processes based on resin transfer molding or resin infusion is seen as animportant factor for controlling costs ([1]).

1

Choosing a structural design (which includes materials) and a manu-facturing process are coupled decisions ([2]). While early composite designstudies simplified the analysis by separating structural design ([3]) and pro-cess tuning ([4]), in the last decade cost has increasingly been considered atan early stage.

A series of works focus on tuning a particular process, e.g., Resin TransferMolding (RTM), and sizing the structure simultaneously ([5, 6]).

Other contributions estimate cost in terms of structural complexity, in-dependent of the process. In [7, 8], ease of prepreg layup is considered as oneof the design criteria for general structures along with strength. In [9], thecomplexity depends on the geometry of ribs and spars of a composite box.

Finally, in [10] and [11, 12], different manufacturing routes are comparedusing cost models.

In terms of optimization methods, most of the aforementioned works relyon enumeration ([10, 5]) or problem decomposition and local searches ([7,12]). Some attempts have been made at using more robust, but numericallymore expensive, global optimization algorithms (e.g., genetic algorithms in[4, 6]). Also, because many criteria need to be traded off against one another,multi-criteria approaches have appeared in [12, 9].

This article presents a methodology to study the effects of taking into con-sideration the process at early design stages in a numerically well-foundedfashion. A simple, yet common, scenario is considered: a rectangular plate ismanufactured by resin transfer molding and compression (RTCM). Injectioncan be flow rate or pressure controlled and compression can follow or be simul-taneous with injection, which means that there are four processes to study.Process criteria are maximum mold pressure and injection time. They affecttooling quality and time to market. They are therefore partly representativeof manufacturing cost. Structural criteria are stiffness and buckling. Underthese assumptions, semi-analytical models of the process and structure areavailable. These models are numerically inexpensive, which enables the use ofa global optimization algorithm. The process parameters and the ply orienta-tions are the design variables. The effects of required structural performanceand maximumallowable mold pressure on optimal processes and designs are characterizedby solving coupled optimization problems.

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2 Design formulation

2.1 Structural and RTCM couplings in composites

When considering the RTCM process, engineers are typically concerned withtuning parameters such as injection pressure/flow rate, temperature, gate/ventspositions and preform characteristics. The criteria they observe are the in-jection time, the maximum mold pressure, the temperature distribution, pre-form damage, void contents and degree of curing. At the same time, struc-tural designers specify materials, geometry, fiber volumes, fiber orientationsand ply thicknesses to achieve minimum mass and specified strength, stiff-ness, damage or other failure criteria. An overview of process and structurevariables and criteria involved in composites engineering is given in Fig. 1.While process and structure engineering are traditionally separate activities,the central part of the Figure shows variables and criteria that are commonto both fields: preform characteristics, void content, polymerization and pre-form damage couple manufacturing and the final structure.

A method will shortly be introduced that estimates the consequences ofaccounting for these couplings in composite laminates. A channel flow modelof the RTCM process ([13]) is used. Among all the RTCM processes that aredescribed in Appendix A, four are compared here :

RTCM-P A successive two step process, starting with a pressure imposedinjection, at P constant, followed by compression at a constant rate,RC . IP % of the laminate length is filled only by injection before com-pression starts, i.e., compression may be applied before the laminate isfully infiltrated.

RTCM-F Same as above, except that injection is performed at a constantflow rate RF .

RTCM-SP Pressure controlled injection and compression are simultaneous.

RTCM-SF Flow rate controlled injection and compression are simultane-ous.

Note that these process models assume a closed mold, i.e., no resin canbe vented in the compression phase. In this case, when IP = 100 % orRC = 0 m/s, the RTCM process degenerates into a RTM process.

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2.2 Accounting for process and structure in the opti-mization problem formulation

The process-structure couplings first take place in the physical model of thesystem that is being optimized. They need to be taken into account furtherin the optimization problem formulation. This is achieved when choosingdesign variables and criteria.

There are two possible formulations for a coupled design problem. Onecan solve a multi-objective optimization problem (e.g., [12, 9]),

minx∈S f1(x) ,and minx∈S f2(x) ,. . .and minx∈S fm+1(x) ,

(1)

where fi are the criteria, x is a vector of design variables and S is the designspace. Alternatively, a constrained single objective problem is formulated bytransforming some criteria into inequalities, gi,

minx∈S f(x) ,and g1(x) ≤ 0 ,. . .and gm(x) ≤ 0 .

(2)

With respect to a criterion, a constraint requires an additional allowablevalue. As an example, the criterion associated with the maximum pressureduring injection, Pmax, is

minPmax , (3)

while the constraint is

satisfy Pmax ≤ P allowmax . (4)

The multi-objective approach, which does not need a priori allowable values,is the most general formulation. It has many – eventually, an infinite numberof – solutions, called the Pareto set, which is all the possible compromisesthat can be struck between the criteria. However, from a numerical pointof view, solving the multi-objective problem (1) is much more complex thansolving the constrained single objective problem (2). State-of-the-art multi-objective algorithms are evolutionary algorithms (see [14] for a review) which

4

have a slower convergence than other single objective algorithms such as theGlobalized and Bounded Nelder-Mead algorithm (see Section 2.4 and [15]).Furthermore, the set of solutions need to be post-processed in order to decidewhich compromise should be manufactured.

In the current work, a complete multi-objective optimization is replacedwith a series of four constrained single objective problems: a process onlyproblem, (PO), a structure only problem, (SO), a coupled problem with lowperforming structure, (CL), and a coupled problem with optimal structure(solution of the structure only problem), (CO). Because the series is com-posed of constrained single objective problems, numerical resolution is eased.Extreme regions of the Pareto set are located in the process and structureonly problems. The solution of the structure only problem is an indicationof which allowable values can be achieved in the coupled problems. From adidactic point of view, comparing the solutions of the four problems showshow simultaneous consideration of the process and the structure affects thefinal design and its manufacturing.

2.3 Detailed design problems formulations

The series of four optimizations are repeated for the four processes RTCM-P,RTCM-F, RTCM-SP and RTCM-SF. In all processes, imposed pressures, flowrates, and compression rates are time constants (but optimization variables).

Although the proposed method can be applied to any laminate, symmetricand balanced laminates of eight layers are considered, so there are two designvariables to describe the stacking sequence, θ1 and θ2. The θ’s are zero whenthe fibers are aligned with the resin flow. I denotes the injection imposedcondition, that is, either a pressure P or a flow rate RF . All design variablesare continuous.

There are five design variables for processes that have successive injectionand compression,

x =

({PRF

, RC , IP , θ1, θ2

), (5)

and four design variables when injection and compression are simultaneoussince IP has no meaning in this case,

x =

({PRF

, RC , θ1, θ2

). (6)

5

The two process design criteria are the total injection time, TP , and themaximum mold pressure, Pmax. Both measures are representative of themanufacturing cost. TP is a small part of the total production time whichis dominated by the curing time. In practice however, the injection time isthe main degree of freedom of the production time since the resin choice islimited. Pmax is an indication of the tooling price. High Pmax implies heavymetal molds while a low Pmax suggests that a composite mold (like in RTM-light, cf. [16]) may be sufficient. Note that when injection and compressionare successive, TP is the total time of the two phases. The models usedto calculate TP and Pmax are presented in Appendix A. In the RTCM-SPprocess, all combinations of pressure and compression rate are not valid. Thepressure in the mold may become larger than the injection pressure, whichcauses an inversion of the resin flow at time Tinv. This scenario is avoided bydefining an inversion time constraint

TP ≤ Tinv , (7)

which is added to all optimization problems involving the RTCM-SP process.Structural design criteria concern the transverse effective stiffness Ey, theshear effective stiffness Gxy and the buckling load factor λ. Calculations ofthe structural criteria are based on the Classical Lamination Theory and areoutlined in Appendix B. Ex, the longitudinal effective stiffness, may also bean important structural property. Yet, Ex doesn’t need to be included herebecause it is redundant with the process criteria, TP and Pmax: Ex, like TPand Pmax, drives the fiber orientation to 0◦.

The process only problem, (PO), is

(PO)

{minI,RC ,IP ,θ1,θ2 TP ,such that Pmax ≤ P allow

max ,(8)

where P allowmax is a user-defined allowed maximum mold pressure. More gener-

ally, allowable values are denoted with the allow superscript. The structureonly problem, (SO), is

(SO)

maxI,RC ,IP ,θ1,θ2 Ey ,such that Gxy ≥ G∗xy ,and λ ≥ λ∗ .

(9)

Let E∗y , G∗xy and λ∗ denote the structural properties at a solution of the (SO)

problem. A first coupled process-structure problem, where the structure has

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a low quality, is

(CL)

minI,RC ,IP ,θ1,θ2 TP ,such that Pmax ≤ P allow

max ,Ey ≥ Eallow

y ,Gxy ≥ Gallow

xy andλ ≥ 1 .

(10)

This problem has a low structural quality because the allowable values inthe structural constraints are chosen such that Eallow

y < E∗y , Gallowxy < G∗xy

and λallow < λ∗. Finally, the coupled process-structure problem with anoptimal structure, (CO), has the same expression as (CL), but the allowablevalues of the structural criteria correspond to the solution of the structureonly problem, i.e., Eallow

y = E∗y , Gallowxy = G∗xy and λallow = λ∗.

2.4 The Globalized and Bounded Nelder-Mead opti-mization algorithm

The constrained single objective optimization problems are dealt with bythe Globalized and Bounded Nelder-Mead algorithm ([15]), GBNM. Thisalgorithm is based on repeated direct Nelder-Mead (sometimes also calledsimplex, [17]) searches, which work on continuous variables, do not need gra-dient information and, individually, converge faster than probabilistic globalsearch methods such as evolutionary algorithms ([18]). The method is madeglobal by restarting local searches based on a probability density which bi-ases new local searches towards unexplored regions of the design space S.Restarts are stopped when a user specified maximum number of analyses,Nmax, has been reached. If the maximum number of analyses is large enoughand the problem has several local optima, the GBNM algorithm typicallyfinds many of them. Note that degeneracy cases of the Nelder-Mead meth-ods are detected and induce re-initialization of the simplex. Additionally,the GBNM algorithm copes with bounds on the variables by projection andgeneral non-linear constraints by an adaptive linear penalty scheme.

3 Numerical applications

A plate of length a = 0.5 m and width b = 0.1 m subject to a bi-axialin-plane load, Nx = 200000 N , Ny = 20000 N , is considered. Its fiber and

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resin properties are : E1,f = 80. 109 Pa, E2,f = 80. 109 Pa, ν12,f = 0.22,G12,f = 35.2 109 Pa, Em = 3.45 109 Pa, νm = 0.3. The initialply thickness (before compression) is hi(0) = 1 mm with a fiber volumefraction vf,i(0) = 0.4, i = 1, 8. The resin viscosity is µ = 0.16 Pa.s. µ isassumed here to be a time constant, which is valid if the curing time is largerthan the injection time. The coefficients of the permeability relations (cf.Equation (18)) are A1 = 1.7 10−12 m2, B1 = − 4.12, A2 = 6.8 10−14 m2

and B2 = − 5.64.

3.1 Parametric study

Before turning to a complete optimization of the system, a parametric studyis performed that gives an insight into the design variables and criteria rela-tionship. To simplify the interpretation, the set of optimization variables ofEquation (5) or (6) is replaced with two variables, that are varied in turn:the fiber orientation θ of a ((±θ)2)s laminate, and the injection length beforecompression starts.

For comparison purposes in the parametric study, the imposed pressureand flow rates are chosen equivalent on average, i.e., they satisfy

RF =1

2µH(0)PKx , (11)

where Kx is the average longitudinal permeability (cf. Appendix A.3) andH(0) is the initial total thickness. The process where injection is pressurecontrolled and simultaneous with compression is not studied here because un-der the above conditions and a compression rate RC = 1. 10−6 m/s, inversionwill always occur.

In Figure 2, the variation of the design criteria as a function of the fiberorientation θ is plotted. The following observations can be done:

• The injection time (top-left plot) is independent of the fiber orienta-tion when the flow rate is imposed, while it increases when pressureis imposed and the fibers are disoriented with respect to the injectiondirection (the permeability decreases).

• The maximum pressure during manufacturing (top-right plot) increaseswhen the fibers are disoriented. The augmentation is drastic wheninjection flow rate is imposed and compression is simultaneous. The

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rise is much higher when flow rate is imposed than when pressure isimposed.

• The simultaneous process, with the given settings, yields a structurethat is thicker than that of successive processes. In other words, thefinal fiber volume fraction, Vf (TP ), is lower. This explains why in-planestiffnesses are lower in the simultaneous process while the buckling loadfactor is larger. Indeed, the in-plane stiffnesses increase linearly withVf ≡ (1/thickness) while the buckling load increases with (thickness3).In terms of buckling, it is more efficient to sacrifice fiber fraction for abetter flexural strength. Note that this observation is case dependent:for a sufficiently higher compression rate or a lower IP or a lower RF ,the simultaneous process yields thinner laminates than the successiveprocesses do.

• It is confirmed that the optimal fiber orientation is 90◦ for Ey and 45◦

for Gxy and buckling.

• The successive processes, with the flow rate or the pressure imposed,produce structures with the same final thicknesses, therefore the samestructural criteria. Indeed, the final thickness depends only on thecompression phase, which is the same in both cases (RC and IP areidentical).

Design criteria are represented as a function of the percentage of platelength filled by injection before compression starts, IP , in Figure 3 under thesame material and process conditions as in Figure 2. The following simplecomments can be made:

• The compression rate is slow with respect to the injection flow rate orpressure, so the injection time decreases with IP . If the compressionrate were higher, opposite conclusions would be drawn.

• For successive processes, the maximum pressure decreases with IP aslong as it occurs during compression. Reciprocally, when maximumpressure occurs during injection, it increases with IP . The top right plotof Figure 3, for RTCM-F and IP beyond 45% gives such an example.

The results of the parametric study are specific to the material chosenand process tuning. No monotony in the criteria variation as a function of the

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variables exists. In order to properly account for all parameter combinations,it is necessary to optimize the system with a global non-linear algorithm.

3.2 Optimization results

The GBNM algorithm (see paragraph 2.4) is applied to 32 optimization prob-lems. Each search is Nmax = 80000 analyses long. Depending on the process,the average CPU time for one analysis ranges from 0.02 to 0.2 s on a 800MHzI686 IntelTM Pentium processor. The 32 problems correspond to the 4 formu-lations, process only (PO), structure only (SO), coupled low structural per-formance (CL) and coupled optimal structure (CO), repeated with the fourprocesses (RTCM-P, RTCM-F, RTCM-SP, RTCM-SF) and two maximumpressures in the mold, P allow

max = 10.105 and 6.105 Pa, respectively. The stack-ing sequence considered is (±θ1/ ± θ2)s. The design variables are boundedas follows : 0.5 105 ≤ P ≤ 6. 105 Pa, 0.5 10−7 ≤ RF ≤ 100. 10−7 m3/s,0. ≤ IP ≤ 100. %, 0. ≤ RC ≤ 0.4 10−3 m/s, 0. ≤ θ ≤ 90. ◦. Allowablecriteria values are Eallow

y = 15.109 Pa, Gallowxy = 5.109 Pa and λallow = 1.

in (CL). Optimal values are Gxy = 6.89 GPa and λ∗ = 1.3 in (SO).By definition, mechanical allowable values in (CO) stem from the solution ofthe (SO) problem (see Paragraph 2.3). The results of the optimizations aregiven in Tables 1 to 5.

The principal significance of these results consist in the numerical solu-tions to the coupled process-structure problems, (CL) and (CO).

(CL) problem : at P allowmax = 10.105 Pa, RTCM-P is optimal with an injec-

tion time of 326 s. Active constraints are Pmax and Ey. RTCM-F andRTCM-SP are second best processes with a time of 390 s. It should benoted that RTCM-SP is a gentle process because it results in a lowermold pressure (6.105 Pa maximum). If P allow

max is decreased to 6.105 Pa,a pressure controlled injection without compression is optimal in 391 s.For low mold pressure, RTM is the optimal process, and both RTCM-Pand RTCM-SP can degenerate into it.

(CO) problem : RTCM-P is again the optimal process for both P allowmax = 10.105

and 6.105 Pa with injection times of 785 and 912 s, respectively. Allconstraints are active (structural constraints are active by constructionof the (CO) problem).

Comparing (CL) with (CO) results shows that high structural perfor-mance more than doubles the injection time. In other words, when aiming

10

at low structural performance, an uncoupled design procedure where, first,the layup is optimized and, second, the process is tuned, doubles the injectiontime with respect to a coupled procedure, (CL).

Solutions of the process only and structure only problems are familiar tomany experts. Yet, in order to be comprehensive, they are now commented.Secondarily, these classical solutions validate the numerical optimizers im-plemented.

Process only problem : fibers are oriented along the resin flow, θ1 =θ2 = 0. Compression is useful to successive processes, except RTCM-P at low P allow

max . Compression never speeds up simultaneous processesbecause of the associated loss of permeability.

Structure only : loss of solution uniqueness for all processes, the struc-ture only problems (SO) have an infinite number of solutions, corre-sponding to all injection and compression settings that produce thesame final thickness. These solutions are not equivalent in terms ofinjection time or mold pressure, but these criteria are not consideredin (SO). It can be seen for the RTCM-P process in Table 1 on the(SO) line. For RTCM-P and a fixed IP , any combination of P and RC

results in the same final thickness, which is a resin conservation prop-erty. Indeed, IP sets the amount of resin injected in the mold whichsubsequently determines how much compression is needed to fill theplate.

Finally, when interpreting the optimization results in terms of processfeatures, the following well-known traits underlie the solutions.

Compression : compression improves in-plane structural stiffnesses and re-duces injection times at the expense of larger mold pressures. There-fore, when high structural performance is not required and mold pres-sure should remain low ((CL) problem, P allow

max = 6.105 Pa), all optimalprocesses but RTCM-F have no compression. In the simultaneous pro-cesses, since compression induces fast mold pressure growth (it reducespermeability), it is absent in all simultaneous processes for low struc-tural quality. In the (CO) formulations, compression is necessary toachieve high structural performance.

Flow control : flow controlled injection is not optimal in our numericalapplications because, for an equivalent injection time, it yields higher

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mold pressure than pressure controlled injection. RTCM-F optimalsettings all use compression in a way such that P allow

max is reached duringboth injection and compression steps. Nevertheless, flow controlledprocesses regain competitiveness for shorter plates.

Simultaneous processes : RTCM-SP and RTCM-SF are not optimal forthe problems considered here. However, for low mold pressures andhigh structural properties, RTCM-SP has a strong potential. WhenP allow

max drops from 10.105 Pa to 6.105 Pa, its time excess with respectto the optimal process shrinks from to 20 to 9%.

4 Concluding remarks

The effects of accounting, at the first design stages, for the couplings betweenprocess and structure have been investigated in the case of an injected andcompressed laminated plate.

The study is based on four design formulations, two where the couplingsare ignored and either the structure or the process are optimized, and twocoupled formulations aiming at obtaining low and high structural perfor-mances, respectively. Four processes, which have pressure or flow controlledinjection and simultaneous or successive injection and compression phases arecompared. Solutions analyses is based on semi-analytical process and struc-ture models, whose numerical efficiency authorizes a global optimization bythe GBNM algorithm.

The optimizations exhibit the fundamental trade-offs that exist betweeninjection time, maximum mold pressure, structural in-plane and flexural stiff-nesses. For example, the manufacturing of a low performance laminate bythe optimal RTCM-P process needs an extra minute when the maximumpressure decreases from 10 to 6.105 Pa. If a high structural performance istargetted, the injection time increases by two minutes.

In terms of perspectives, this work lays the foundations for an approachto select, based on a final structure, a resin molding techniques from a widerpanel of processes (Advanced RTM, Liquid Resin Infusion, Resin Film Infu-sion, . . . ).

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A RTCM process analysis

A.1 A family of RTCM processes

The generic resin transfer and compression molding scenario considered inthis study is the case of a rectangular plate injected on one side and uniformlycompressed (cf. Figure 4). If one further assumes a channel-like flow, theanalysis is one-dimensional.

Many combinations of the RTCM process settings are possible. Theymake up a family of processes, of which four are optimized in the bodyof this article : the injection is either pressure or flow rate controlled atconstant P or RF respectively. Compression eventually occurs at a constantrate, RC . Compression and injection may be simultaneous (the RTCM-S. . . processes) or successive. During pure compression, a vent may be openedat x = 0, allowing resin to flow out of the mold, or not. Finally, constant ortime varying viscosities µ can be accounted for. All these RTCM processesare analyzed in [13], resulting in 16 models to calculate injection time andmaximum mold pressure.

A.2 Calculation of injection time and mold pressure

The one-dimensional flow of Figure 4 can be analyzed using Darcy’s law anda continuity equation which together yield ([19])

∇ ·(−Kx

µ∇p)

= − RC

H, (12)

whereKx is the macroscopic longitudinal permeability (more details are givenin Paragraph A.3), µ is the resin viscosity, p the resin pressure, and H thetotal plate thickness. IfH, µ, Kx and Vf depend only on time t, Equation (12)can be integrated for p(x, t). The pressure field is quadratic if compression isapplied, linear otherwise. The differential equation describing the resin frontposition also stems from Darcy’s law and is

dxFdt

= − 1

H(1− Vf )

(RCxF −

RF

b

), (13)

where Vf is the average fiber volume and b is the plate width. A total thick-ness increment ∆H is shared among the N plies by looking at the stacking

16

sequence as springs of stiffness E2,i in series. If ∆H is applied to the laminate,the i-th ply thickness, hi, changes by

∆hi =hi∆H

E2,i

∑Nj=1 hj/E2,j

, (14)

where E2,i is the i-th ply transverse Young’s modulus. Injection and com-pression times are calculated by integrating Equation (13). For example, ifthe injection is pressure controlled at P , the front position follows,

H(1− Vf )xF = 2

∫ t

ti

HKxP

µds+ (1− Vf (0))H(0)xF (ti)

2 . (15)

In the case of the pressure controlled RTM with simultaneous compression(RTCM-SP), the process time TP is solution of the non-linear equation ([13])

a2 =2P

(RCTP + (1− Vf (0))H(0))(H(0)I1(TP ) +RCI2(TP )) , (16)

where a is the plate length and

I1(t) =

∫ t

0

Kx

µds , I2(t) =

∫ t

0

sKx

µds . (17)

Equation (16) is solved by numerical dichotomy.

A.3 Permeability estimation

An average plate longitudinal permeability Kx is estimated from the pliespermeability tensors. Firstly, permeabilities in the principal direction of thei-th ply are calculated from the fiber volume fraction using empirical laws

k1,i = A1vB1f,i , k2,i = A2v

B2f,i . (18)

Next, the ply longitudinal permeability of the i-th ply is obtained by rotatingthe permeability tensor in the principal direction along the x-axis,

kx,i =k1,i + k2,i

2+k1,i − k2,i

2cos(2θi) , (19)

where θi is the i-th ply orientation. Finally, ply longitudinal permeabilitiesare averaged based on the ply thicknesses,

Kx =1

H

N∑i=1

hikx,i . (20)

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B Structure analysis

B.1 Micromechanical analysis

Including micromechanical relations in our analysis allows us to study theeffect of fiber and matrix properties changes, first on the ply properties,and next on laminate properties through the Classical Lamination Theoryoutlined in Appendix B.2. Ply mechanical properties are calculated frommicromechanical data according to the relations given in [20], where thefibers are assumed to be transversly isotropic. Denoting by f and m fiber andmatrix properties, respectively, by E Young’s moduli, by G shear stiffness,by ν Poisson’s ratio and by vf the ply fiber volume (a simplification of thevf,i notation of the text for the i-th ply), the micro-meso relationships are,

E1 = vfE1,f + (1− vf )Em , (21)

E2 =Em

1−√vf (1− Em/E2,f ), (22)

G12 =Gm

1−√vf (1−Gm/G12,f ), (23)

ν12 = vfν12,f + (1− vf )νm . (24)

B.2 Classical Lamination Theory

Classical Lamination Theory ([21]) is used to calculate laminate proper-ties, i.e., we apply Kirchhoff plate theory, and neglect through-the-thicknessstresses and strains. Under those conditions, the forces N and moments Mper unit length of the cross section acting on a laminate (cf. Figure 5) arelinked to the strains ε and curvatures κ through,{

NM

}=

{(Nx Ny Nxy)

T

(Mx My Mxy)T

}=

[A BB D

]{εκ

}=

[A BB D

]{(εx εy εxy)

T

(κx κy κxy)T

}.

(25)

The coefficients of the 3 × 3 extensional, coupling and flexural stiffness matri-ces, A, B, D, respectively, are functions of the material properties, the fiberorientations in each ply, and the position of ply interfaces. The coefficients ofthe A, B and D matrices can be written as linear combinations of material

18

invariants Ui’s and lamination parameters Vi’s. As the names suggest, thematerial invariants depend only on the unidirectional material properties E1,E2, G12 and ν12 of Equations (21-24), while the lamination parameters areonly a function of the ply orientations and thicknesses. Detailed expressionsof A, B and D using lamination parameters can be found in [22].

The stiffness matrix of Equation (25) can be inverted, which yields thelaminate compliance matrix,[

A BB D

]−1

=

[a bbT d

]. (26)

The effective Young’s moduli of the laminate in the longitudinal, and trans-verse direction, Ex, Ey, and shear modulus Gxy respectively, are expressedas,

Ex =a11

H, Ey =

a22

H, Gxy =

a66

H, (27)

where the index 6 points to the third row/column of the relevant matrix.

B.3 Buckling analysis

A closed-form solution exists to predict the onset of buckling under the fol-lowing conditions: the plate is flat, symmetric (B = 0), quasi-orthotropic(D16 ≈ D26 ≈ 0), simply-supported and subject to bi-axial in-plane loads(Nx and Ny only). The plate buckles with m and n half-waves in the x andy directions respectively, when the load reaches (λmnNx, λmnNy), with

λmn = − π2[D11(m/a)4 + 2(D12 + 2D66)(m/a)2(n/b)2 +D22(n/b)4]

(m/a)2Nx + (n/b)2Ny

. (28)

The critical load factor, λ, corresponds to the combination of m and n thatminimizes λmn. If 0 < λ < 1, the plate fails under the (Nx, Ny) loading.

19

� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �

� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �

� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �

� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �

� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �

� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �

preform texturematrix (resin viscosity)

gate/vents positionstemperatureinjection pressure / flow rate

vari

able

spr

oces

s

vari

able

sst

ruct

ure

fiber volumes, orientationsply thicknesses

fiber and matrix stiffnessfiber and matrix density

process timemax. pressuremax. temperature

void content

proc

ess

crite

ria

COUPLINGS

structuralengineering

processengineering

eigenfrequenciesbuckling factormass dilatation coefficientsstrengthstiffness

crite

ria

stru

ctur

e

preform damagedegree of curing

Figure 1: Process and structure couplings.

20

0

500

1000

1500

2000

2500

3000

3500

0 10 20 30 40 50 60 70 80 90

pro

cess

tim

e (s

)

theta

0

1e+06

2e+06

3e+06

4e+06

5e+06

6e+06

7e+06

0 10 20 30 40 50 60 70 80 90

max

. pre

ssur

e (P

a)

�

theta

5e+09

1e+10

1.5e+10

2e+10

2.5e+10

3e+10

3.5e+10

4e+10

0 10 20 30 40 50 60 70 80 90

Ey

(Pa)

theta

3e+094e+095e+096e+097e+098e+099e+091e+10

1.1e+10

0 10 20 30 40 50 60 70 80 90

Gxy

(Pa

)

�

theta

0.80.9

11.11.21.31.41.51.61.7

0 10 20 30 40 50 60 70 80 90

buc

klin

g lo

ad f

acto

r

theta

Figure 2: Design criteria vs. θ for 3 processes, RTCM-P (+), RTCM-F(×) and RTCM-SF (◦), ((±θ◦)2)s, P = 3. 105 Pa, RF = 5.5 10−7 m3/s,IP = 80%, RC = 1. 10−6 m/s.

21

0500

100015002000250030003500400045005000

0 10 20 30 40 50 60 70 80 90 100

pro

cess

tim

e (s

)

percent injection

02e+064e+066e+068e+061e+07

1.2e+071.4e+071.6e+071.8e+07

0 10 20 30 40 50 60 70 80 90100

max

. pre

ssur

e (P

a)

�

percent injection

2e+10

3e+10

4e+10

5e+10

6e+10

7e+10

8e+10

0 10 20 30 40 50 60 70 80 90 100

Ey

(Pa)

percent injection

5e+09

1e+10

1.5e+10

2e+10

2.5e+10

3e+10

3.5e+10

0 10 20 30 40 50 60 70 80 90100

Gxy

(Pa

)

�

percent injection

0.20.40.60.8

11.21.41.61.8

0 10 20 30 40 50 60 70 80 90 100

buc

klin

g lo

ad f

acto

r

percent injection

Figure 3: Design criteria vs. percentage of injection length before compres-sion starts for 3 processes, RTCM-P (+), RTCM-F (×) and RTCM-SF (◦),(±54◦/± 90◦)s, P = 3. 105 Pa, RF = 5.5 10−7 m3/s, RC = 1. 10−6 m/s.

22

��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

x F

RF

P,

RC

������������������������������������������������������������������������������������������������������������������������������������������������������������������������

���������������������������������������������������������������������������������������������������������������������������������������������������

��������������������������������������������������

���������������������������������������������

a

H

0

z

xinje

ctio

ncompression

Figure 4: RTCM scenario considered.

Nx

Mxy

NxyMx

Mxy

Ny

My

NxyNy

Nx

MxyMy

Mx Mxy

θ

Figure 5: Laminate conventions.

23

Nomenclature∗ , allow optimal , allowable valuea , b plate length and widthA1, A2, B1, B2 permeability law parameters(CL) coupled low structural performance optimization problem(CO) coupled optimal structure optimization problemE1,f , E2,f longitudinal and transverse fiber Young’s moduliEm resin Young’s modulusEy transverse effective stiffnessG12,f fiber shear stiffnessGxy shear effective stiffnesshi(t) i-th ply thickness at time tH(t) laminate total thickness at time tIP % of laminate length filled before compressionKx average longitudinal permeabilityλ buckling load factorµ resin viscosityν12,f , νm fiber and resin Poisson’s ratioNx , Ny longitudinal, transverse in-plane loadP injection pressure(PO) process only optimization problemPmax maximum mold pressureRC compression rateRF injection flow rate(SO) structure only optimization problemθ , θi fiber orientation (in i-th set of plies)Tivn inversion time (RTCM-SP process only)TP total injection timevf,i(t) i-th ply fiber volume fraction at time tVf (t) average fiber volume fraction at time tx generic vector of optimization variables

24

PRC

I Pθ 1

θ 2TP

Pm

ax

Ey

λGxy

Hf

(105Pa)

(10−

5m/s

)(%

)(◦

)(◦

)(s

)(1

05Pa)

(GPa)

(GPa)

(mm

)(PO

)6.

1.70

282.8

80.

0.23

3.10.

9.55

0.87

3.71

7.17

8(SO

)†

†81.7

554.4

690.

[785.,∞

[[P,∞

[27.6

41.

36.

897.

125

(CL

)6.

1.14

781.3

154.7

10.

326.

10.

15.

1.31

6.89

7.10

3(CO

)6.

0.46

181.7

554.4

690.

785.

10.

27.6

41.

36.

897.

125

Tab

le1:

(PO

),(SO

),(CL

)an

d(CO

)so

luti

ons

for

the

RT

CM

-Ppro

cess

,P

all

ow

max

=10.1

05Pa.†

:al

lP

andRC

are

opti

mal

,as

soci

ated

pro

cess

crit

eria

are

give

nas

inte

rval

s.

RF

orP

RC

I Pθ 1

θ 2TP

Pm

ax

Ey

λGxy

Hf

(10−

7m

3/s

(10−

5m/s

)(%

)(◦

)(◦

)(s

)(1

05Pa)

(GPa)

(GPa)

(mm

)or

105Pa)

RTCM−F

6.27

1.12

881.9

658.7

40.

390.

10.

16.3

61.

296.

507.

134

RTCM−P

6.1.

147

81.3

154.7

10.

326.

10.

15.

1.31

6.89

7.10

3RTCM−SF

5.11

0.†

59.3

50.

470.

10.

15.

1.63

5.82

8.RTCM−SP

6.0.

†59.3

50.

391.

6.15.

1.63

5.82

8.

Tab

le2:

Cou

ple

dop

tim

izat

ion

wit

hlo

wst

ruct

ura

lp

erfo

rman

ce,(CL

),fo

ral

lpro

cess

es,P

all

ow

max

=10.1

05Pa.

†:

Var

iable

not

rele

vant

toth

epro

cess

.

25

RF

orP

RC

I Pθ 1

θ 2TP

Pm

ax

Ey

λGxy

Hf

(10−

7m

3/s

(10−

5m/s

)(%

)(◦

)(◦

)(s

)(1

05Pa)

(GPa)

(GPa)

(mm

)or

105Pa)

RTCM−F

2.75

0.46

081.7

554.4

690.

905.

10.

27.6

41.

306.

897.

125

RTCM−P

6.0.

460

81.7

554.4

690.

785.

10.

27.6

41.

306.

897.

125

RTCM−SF

1.03

0.04

6†

54.4

690.

1894.

10.

27.6

41.

306.

897.

125

RTCM−SP

6.0.

088

†54.4

690.

992.

6.27.6

41.

306.

897.

125

Tab

le3:

Cou

ple

dop

tim

izat

ion

wit

hop

tim

um

stru

cture

,(CO

),fo

ral

lpro

cess

es,P

all

ow

max

=10.1

05Pa.†

:V

aria

ble

not

rele

vant

toth

epro

cess

.

RF

orP

RC

I Pθ 1

θ 2TP

Pm

ax

Ey

λGxy

Hf

(10−

7m

3/s

(10−

5m/s

)(%

)(◦

)(◦

)(s

)(1

05Pa)

(GPa)

(GPa)

(mm

)or

105Pa)

RTCM−F

3.80

0.64

380.6

559.3

50.

654.

6.16.7

11.

256.

497.

071

RTCM−P

6.0.‡

100.

0059.3

50.

391.

6.15.

1.63

5.82

8.RTCM−SF

3.06

0.†

59.3

50.

783.

6.15.

1.63

5.82

8.RTCM−SP

6.0.

†59.3

50.

391.

6.15.

1.63

5.82

8.

Tab

le4:

Cou

ple

dop

tim

izat

ion

wit

hlo

wst

ruct

ura

lp

erfo

rman

ce,

(CL

),fo

ral

lpro

cess

es,P

all

ow

max

=6.

105Pa.

†:

Var

iable

not

rele

vant

toth

epro

cess

.‡

:RC

arbit

rari

lyse

tto

0b

ecau

seth

ere

isno

com

pre

ssio

n(IP

=10

0).

26

RF

orP

RC

I Pθ 1

θ 2TP

Pm

ax

Ey

λGxy

Hf

(10−

7m

3/s

(10−

5m/s

)(%

)(◦

)(◦

)(s

)(1

05Pa)

(GPa)

(GPa)

(mm

)or

105Pa)

RTCM−F

1.65

0.27

681.7

654.4

690.

1508.

6.27.6

41.

306.

897.

125

RTCM−P

6.0.

276

81.7

654.4

690.

912.

6.27.6

41.

306.

897.

125

RTCM−SF

0.62

10.

028

†54.4

690.

3156.

6.27.6

41.

306.

897.

125

RTCM−SP

6.0.

088

†54.4

690.

992.

6.27.6

41.

306.

897.

125

Tab

le5:

Cou

ple

dop

tim

izat

ion

wit

hop

tim

um

stru

cture

,(CO

),fo

ral

lpro

cess

es,P

all

ow

max

=6.

105Pa.†

:V

aria

ble

not

rele

vant

toth

epro

cess

.

27