A Simulation App based on reduced order modeling for manufacturing optimization of composite outlet guide vanes
 Jose Vicente Aguado^{1},
 Domenico Borzacchiello^{1},
 Chady Ghnatios^{2},
 François Lebel^{3},
 Ram Upadhyay^{3},
 Christophe Binetruy^{4} and
 Francisco Chinesta^{1}Email author
DOI: 10.1186/s403230170087y
© The Author(s) 2017
Received: 21 March 2016
Accepted: 4 January 2017
Published: 17 January 2017
Abstract
Composites manufacturing processes usually involve multiscale models in both space and time, highly nonlinear and anisotropic behaviors, strongly coupled multiphysics and complex geometries. In this framework, the use of simulation for realtime decision making directly in the manufacturing facility is still precluded nowadays, in spite of the impressive progresses reached in numerical analysis and computer science during the last decade. In this paper, a processspecific simulation tool based on reduced order modeling is introduced, the Simulation App. This concept is presented through a practical case involving a multiphysics and coupled problem describing the manufacturing process of a composite outlet guide vane. We show that several manufacturing settings can be simulated in few seconds with the Simulation App, thus enabling fast process optimization. Finally, the advantages over generalpurpose simulation software, in the context of process simulation, are discussed.
Keywords
Simulation App Reduced order modeling Real time simulation Numerical simulation Composites forming processes Curing Consolidation Proper Generalized Decomposition Parametric solutions Computational VademecumsBackground
Efficient simulation of composite manufacturing processes remains even nowadays, in many cases, a challenging issue, mainly when they involve rich 3D behavior, multiphysics and the necessity of solving many scenarios very fast for optimization purposes [1, 2]. Composites manufacturing processes involve many different physics. For instance, impregnation of fibrous reinforcements involves flow models through porous media, combined with the mould compression for ensuring an appropriate degree of consolidation [3, 4]. During consolidation the resin curing kinetics strongly affects its viscosity and therefore the flow conditions [5].
The intimate coupling between the resin flow, the reinforcement permeability, that decreases upon compression of the mould, and the viscosity, that depends on the temperature and the curing degree, can be at the origin of different process defects. Large viscosities imply high pressures that can generate irreversible preform deformations and even local reinforcement displacements creating local inhomogeneity defects. Moreover, residual stresses are another manifestation of this intimate multiphysics coupling [1, 6].
Another important issue encountered in the simulation of composites manufacturing is the one related to the process control and optimization [7–9]. In general, optimization implies the definition of a cost function and the search of the optimum process parameters defining the minimum of that cost function. The procedure starts by a guessed set of process parameters. Then the process is simulated using suitable numerical methods. The solution of the model is the most expensive step of the optimization procedure. As soon as the solution is available, the cost function can be evaluated and its optimality checked. If the chosen parameters do not define a minimum (at least local) of the cost function, the process parameters should be updated and the solution recomputed. The procedure continues until reaching the minimum of the cost function. The solution of the process model is a tricky task that demands important computational resources and usually implies extremely large computing times. Thus, usual optimization procedures are inapplicable under the realtime constraint. The same issues are encountered when dealing with inverse analysis in which material or process parameters are expected to be identified from numerical simulation, by looking for the unknown parameters so that computed fields match the ones measured experimentally.
Until now, the solution consisted in using the more and more powerful computing platforms and techniques for speeding up standard discretization techniques. Appealing alternatives for circumventing, or at least alleviating, these issues lie in the use of reduced order modeling (ROM) [10–12] strategies. ROM is based on the observation that the family of parametric solutions of a given model usually contains much less information than it was originally assumed when the discrete model was built. Proper Orthogonal Decomposition, Proper Generalized Decomposition and Reduced Basis are nowadays widely considered from a fundamental and applicative viewpoints.
Proper Orthogonal Decomposition (POD) is a general technique for extracting the most significant characteristics of a system’s behavior and representing them in a set of “POD basis vectors” [13, 14]. These basis vectors then provide an efficient (typically lowdimensional) representation of the key system behavior, which proves useful in a variety of ways. The most common use is to project the system governing equations onto the reducedorder subspace defined by the POD basis vectors. This yields an explicit POD reduced model that can be solved in place of the original system. The POD basis can also provide a lowdimensional description in which to perform parametric interpolation [15, 16], infill missing or “gappy” data [17], hyperreduced approximations [18, 19], and perform model adaptation. There is an extensive literature on POD showing it has broad application across fields. Some review of POD and its applications can be found in [20, 21] and the references therein.
Another family of ROM techniques is based on the use of a reduced basis constructed by combining a greedy algorithm and “a posteriori” error indicators [22–24]. As for the POD, the Reduced Basis method requires some amount offline work, but then the reduced basis model can be used online for solving different models with control of the solution accuracy, because the availability of error bounds [25]. When the error is unacceptably high, the reduced basis can be enriched by invoking a greedy adaption strategy. Useful review works on the subject are [26, 27].
Techniques based on the use of separated representations are at the heart of the socalled Proper Generalized Decomposition (PGD) methods [28–30]. Such separated representations were considered in computational mechanics for separating space and time in transient solutions [31, 32]. Separated representations were employed for solving multidimensional models suffering the socalled curse of dimensionality [33–35] and in the context of stochastic modeling [36]. Then, they were extended for separating space coordinates making possible the solution of models defined in degenerated domains, e.g. plate and shells [37] as well as for addressing parametric models where model parameters were considered as model extracoordinates, making possible the offline calculation of the parametric solution that can be viewed as a metamodel or a computational vademecum, to be used online for real time simulation, optimization, inverse analysis and simulationbased control [38]. Some applications in the context of composites manufacturing processes were addressed in [39] while the multi physics coupling was successfully achieved in [40].
This work is intended to propose a first approach using reduced order modeling techniques to enhance adaptability of composite manufacturing process to changeable material and process environments through increased parametric modeling capabilities. The process selected is the consolidation and curing of a real part. Consolidation and curing of thermoset preimpregnated fibers (from now on, simply referred as prepregs) involve different physics: heat transfer, compression of fiber beds, resin flow and chemical reaction. Strong couplings exist between these physics and many material parameters come into play. In this paper we combine several different modeling and simulation strategies for the efficient solution of a generic multiphysic and coupled problem. In particular, we propose a ROMbased segregated approach (rather than a monolithic one) for treating each physics separately. This approach allows applying, in each case, the most convenient ROM technique. Then, coupling is made by defining an appropriate parametrization of the coupling variables. A strategy for coupling ROMs of different kind (i.e. reduced solvers and parametric solutions) is also proposed. The integration of all these models constitutes a Simulation App that allows realtime evaluation of any process conditions. The described methodology can be extended and generalized to other processing technologies.
The rest of the paper is organized as follows. “Process description and physics modeling” section presents the manufacturing process as well as the equations describing the physics involved in the manufacturing process of composite outlet guide vanes (OGV). “Simulation based on reduced order modeling” section describes the simulation strategy, including the different simulation modules based on ROM and their coupling. In particular, we recall the main features of the ROM methods that were implemented, emphasizing both the construction of the reduced models in the offline stage and their utilization in the online stage, as a part of the Simulation App. Finally, “A Simulation App for the OGV manufacturing process” section presents how the simulation modules were integrated into a processspecific simulation tool, a Simulation App. We describe the different functionalities of the application, organized by tabs, allowing to the user to perform the usual preprocessing, simulation and visualization operations. The advantages of processspecific simulation tools, such as the Simulation App, over generalpurpose simulation software are discussed, especially in process optimization framework.
Process description and physics modeling
The concept of a Simulation App is presented in this paper through a practical application: the manufacturing process of a composite blade, and more specifically, an outlet guide vane. In this section we are first describing the manufacturing process from a technological point of view. Then we shall present the physics modeling and constitutive behavior for each considered physical phenomena.
Process description
During the forming process, the composite layup is heated by conduction from the top and bottom metallic mold walls and consolidated under press. See Fig. 2 for details. The heat initiates the cure reaction and the applied pressure provides the force needed to drain the excess of resin out of the composite, to consolidate prepreg layers and to reduce voids by compressing the air inside. The temperature raise determines the onset of the cure reaction. Because of the exothermic effects of the curing process and the variability of the kinetic and thermal properties of the resin with the temperature and curing degree, the process is highly nonlinear. Moreover, the resin undergoes a strong rheological modification because its viscosity also depends on the temperature and the degree of cure. Therefore the flow conditions vary continuously with time.
The nominal heating cycle is designed so as to apply a constant temperature of 438 K (330 F) on both top and bottom mold walls. The press closure is initially based on a constant closure rate of 0.1524 mm/min (6 mils/min). As the cure reaction advances, the viscosity increases and so the closing force to be applied by the press increases. When the maximum closing force of the press is attained, the control switches to a forcebased one. The closure rate is therefore adapted so as to keep the closing force constant. A final technological restriction to be taken into account is the socalled hardstop condition, occurring when the cumulated displacement reaches its maximum value, 3.3 mm (130 mils).
Remark 1
(On the simulation of the press control system) Observe that the manufacturing process simulation will be nonlinear due to the just described press control system. If a forcebased control applies, one has to solve a nonlinear problem in order to find out the closure rate that keeps the force constant at its maximum value.
Thermokinetic model
Material parameters considered in the physical modeling of the OGV part manufacturing process
\(k_1\)  \(1.77 \cdot 10^{5}\) s\(^{1}\) 
\(k_2\)  \(9.30 \cdot 10^{2}\) s\(^{1}\) 
\(E_1/R\)  9, 200 K 
\(E_2/R\)  6, 450 K 
n  0.91 
m  0.37 
\(\Delta H\)  570 J/g 
\(\eta _0\)  \(3.10\cdot 10^{12}\) Pa s 
B  10800 K 
C  27 
d  1.77 
\(c_\parallel \)  \( 9.43\cdot 10^{13}\) m\(^2\) 
\(c_\perp \)  \(2.50\cdot 10^{12}\) m\(^2\) 
\(V_{fa}\)  \(78.5\%\) 
\(p_0\)  0.1203 J/gK 
\(p_1\)  0.005 J/gK\(^2\) 
\(q_0\)  0.5905 J/gK 
\(q_1\)  0.0019 J/gK\(^2\) 
\(\rho _\text {resin}\)  1.26 g/cm\(^3\) 
\(\rho _\text {fiber}\)  1.80 g/cm\(^3\) 
\(V_{f0}\)  \(57\% \) 
\(\lambda _{11}\)  5.80 W/mK 
\(\lambda _{22}\)  0.57 W/mK 
\(\lambda _{33}\)  0.57 W/mK 
Consolidation model
Simulation based on reduced order modeling
In this section, we describe the simulation strategy, including the different simulation modules based on ROM techniques and their coupling. In particular, we recall the main features of the ROM methods that were implemented, emphasizing both the construction of the reduced models in the offline stage and their utilization in the online stage, as a part of the Simulation App.
The coupled model described in “Process description and physics modeling” section involves three primary unknown fields, the temperature \(T(\mathbf x, t)\), the pressure \(P(\mathbf x,t)\) and the curing degree \(\alpha (\mathbf x,t)\). The viscosity field, \(\eta (\mathbf x, t)\), can be obtained as a postprocessing of the temperature and curing degree fields. The nonlinearity associated to the coupling can be efficiently addressed by using a semiimplicit incremental time integration, and treating all coupling terms explicitly, therefore allowing for decoupling of the different problems. Thanks to this strategy, we are able to define different simulation modules, described below, one for each physics involved in the manufacturing process. This approach is particularly effective in the context of ROM, because it allows applying the most suitable techniques depending on the nature of the equations.
It is to be noted that domain changes due too the applied compression will be neglected, i.e. the mould thickness reduction remains small enough.
Reduced order modeling methods
The simulation strategy combines three different numerical techniques: (i) the Proper Orthogonal Decomposition (POD), (ii) the Discrete Empirical Interpolation Method (DEIM) and (iii) the Proper Generalized Decomposition (PGD). In this section we summarize the main ingredients of these three techniques.
The Proper Orthogonal Decomposition
POD extracts the most significant components, measured in 2norm, in the solutions of a parametric problem from the analysis of a set of “snapshots” (previously computed solutions of a given problem at different times and for different values of the model parameters) and uses them to approximate the solution for a new set of parameters up to a certain degree of accuracy.
The discrete empirical interpolation method
The Proper Generalized Decomposition
Most of the existing model reduction techniques proceed by extracting a suitable reduced basis and then projecting the problem solution on it. Thus, the reduced basis construction precedes its use in the solution procedure, and one must be careful on the suitability of a particular reduced basis when employed for representing the solution of a particular problem.
This issue disappears if the approximation basis is constructed at the same time that the problem is solved. Thus, each problem has its associated basis in which its solution is expressed. One could consider few terms in its approximation, leading to a reduced representation, or all the terms needed for approximating the solution up to a certain accuracy level. The Proper Generalized Decomposition (PGD) proceeds in this manner.
The thermokinetic simulation module
A POD approach is applied in order to reduce the computational complexity of the thermokinetic simulation. Reduced basis are computed for both primary fields, temperature and curing degree. Other reduced basis are also computed in order to approximate nonlinearities using the DEIM. The training stage, i.e. snapshots generation, and the reduced basis extraction are explained in “Training stage and reduced basis extraction” section. Then, in “Assembling the Reduced order model” section, the ROM for the thermokinetic simulation is assembled. The reduced version of the heat equation is formed by standard Galerkin projection onto the temperature reduced basis. The kinetic equation can be treated in a different manner thanks to its local nature. The computational complexity is reduced in this case by integrating Eq. (1b) only in a wellchosen subset of mesh nodes; in particular, that subset will be computed using DEIM.
Training stage and reduced basis extraction

Initial degree of curing, \(\alpha _0\), that may differ from one OGV part to another depending on the time the prepregs stay out of the freezer before manufacturing.

Initial temperature, \(T_0\), that may change depending on the ambient conditions.

Temperature cycle imposed at both top and bottom walls of the mold, denoted by \(T_t(t)\) and \(T_b(t)\), respectively. In the actual process, however, a fixed temperature is kept all along the process, i.e. the variability at both the beginning and the end can be neglected. In addition, both top and bottom temperatures are the same in practice so we can simply denote \(T_c\equiv T_t = T_b\).

Initial fiber volume fraction, \(V_{f0}\), that may change if the ply stack is modified by adding or removing plies.
The nonlinear terms in the thermokinetic model, namely the specific heat and the curing function, also need a special treatment in order to build an efficient reduced solver. According to “The Discrete Empirical Interpolation Method” section, a reduced basis has to be formed for both specific heat and curing function.
Assembling the reduced order model
Remark 2
(On the approximation of the kinetic rate) As explained in “Training stage and reduced basis extraction” section, the kinetic rate is approximated using the basis for the curing degree, and so in the righthand side of Eq. (22) we use \(\phi _\alpha ^n\). However, it is worth to remark for the sake of clarity that we keep the coefficients \(a_f^n\), which are obviously not the same than \(a_\alpha ^n\).
Recall that coefficients \(a_C^n\) and \(a_f^n\) are computed, at each time, by solving a linear system of size \(N_C\) and \(N_\alpha \), respectively, as explained in “The Discrete Empirical Interpolation Method” section.
Implementation details and results

The Finite Element mesh is composed of 116, 136 HEX8 elements, with 147, 245 nodes.

A firstorder semiimplicit integration scheme was applied. The simulated time is 2 h of process, with 20,000 time steps.

A Preconditioned Conjugate Gradient is used as linear solver.

The typical running time, for a single parameter execution, is 5 h on a 64bit machine with the following specifications: 2.9 GHz Intel Core i5 with 16 Gb 1867 MHz DDR3, running under MacOS X 10.10.5.

A variable order (1–5) Numerical Differentiation Formula (MATLAB ode15s) time integrator was used. The simulated time is 2 h and the number of time steps varies adaptively.

The typical running time is in the range 1–10 seconds depending on the process conditions, using the same machine described previously. It is important to emphasize that the simulation took few seconds to completion instead of the many hours needed when solving the full order problem.

The random access memory (RAM) consumption was measured by running 5 simulations, each one of them for a different set of process parameters selected randomly. The simulation time was set to 1 h. In such conditions, the averaged memory required is 236 Mb. More details about the RAM consumption of the final implementation will be given in “A Simulation App for the OGV manufacturing process” section.

A posteriori validation of the reduced model was carried out by comparing results with fullorder FEM simulations for sets of randomly generated process parameters. The observed relative error measured in the maximum norm is typically of the order of \(10^{5}\).
The consolidation simulation module
The PGD method was applied in order to reduce the computational complexity of the consolidation model. Recall that the consolidation model is nonlinear itself, because the permeability tensor depends on the fiber volume fraction. However, as explained in “Consolidation model” section, the permeability tensor can be assumed uniform, i.e. not varying through the domain. In consequence, observe that the problem could be linearized by simply considering the permeability tensor as an extracoordinate in the PGD framework. A similar approach was already addressed in the context of nonlinear soil dynamics [44]. Hence, in this scenario, we would have a parametric pressure field in the form: \(P(\mathbf {x},\mathbf {K})\). And more importantly, the computation of such parametric would now involve a linear problem. This can be understood by thinking of the parametric solution as being a linear solver inside a fixedpoint nonlinear scheme.
We shall explain in next section how to compute the coefficients \(\mathbf {a}_\eta \) in order to be able to evaluate the parametric solution.
Remark 3
(On the dependence of the pressure field on the closure rate) Simulating the closure cycle of the press requires obtaining the pressure field, for a given viscosity field, permeability tensor (i.e. fiber volume fraction) and an imposed closure rate. However, it is worth to remark that the closure rate does not need to be considered as an extracoordinate of the parametric solution, because of the linearity of the problem to be solved. In other words, the pressure field depends linearly on the closure rate.
Implementation details and results

The same mesh already described in “Implementation details and results” section was used.

The inverse of the viscosity is parametrized using \(N_\eta = 3\) coordinates, corresponding to the first three modes \(\phi ^1_\eta (\mathbf x)\), \(\phi ^2_\eta (\mathbf x)\) and \(\phi ^3_\eta (\mathbf x)\) that can represent the \(95\%\) of the solution.

The FE mesh for the viscosity in the phasespace (\(a_\eta ^1,a_\eta ^2, a_\eta ^3\)) is made of 6, 307 TET4 elements and 1387 nodes.

The FE mesh for the permeability phasespace \(k_{11}, k_{22}, k_{33}\) is made of EDGE2 elements \(1\text {D}\times 1\text {D}\times 1\text {D}\), 100 nodes each.

The PGD convergence criterion was set to \(10^{3}\) on the relative norm of the residual.
Online simulation strategy: simulation modules coupling
 1.
We evaluate both the specific heat and the curing function at the previous time step, i.e. a semiimplicit scheme is used. The reduced coordinates of both specific heat and curing function, denoted by \(\mathbf {a}_C(t_{j1})\) and \(\mathbf {a}_f(t_{j1})\) respectively, are computed using DEIM.
 2.
With those reduced coordinates at hand, the thermokinetic ROM, Eqs. (24) and (26), can be assembled as explained in “The thermokinetic simulation module” section.
 3.
By performing a time increment, the reduced coordinates of both temperature and curing degree can be obtained at time \(t_j\).
 4.
Then, the reduced coordinates of the viscosity, \(\mathbf {a}_\eta \), can be computed by evaluating the chemorheology model and using DEIM.
 5.
With the previous information at hand, the consolidation module can be run. It involves solving a nonlinear problem because of the press control system, see “Process description” section. The nonlinear iterations are indexed by \(\ell \). We start by assuming that the closure rate is the same than in the previous time increment, i.e. \(\dot{U}^{\ell =0}(t_j)= \dot{U}(t_{j1})\). Similarly, we consider \(V_f^{\ell =0}(t_j)=V_f(t_{j1})\).
 6.
Using Eq. (8), the permeability tensor can be computed.
 7.
With the permeability tensor and the reduced coordinates of the viscosity, the parametric PGD solution can be particularized. Recall that this solution has been computed for unitary velocity and so it must be scaled by the actual velocity. We denote by \(\mathbf {P}^\ell (t_j)\) the pressure field at time \(t_j\) and iteration \(\ell \).
 8.
The velocity field can be then computed using Eq. (5). Integrating on the free boundary \(\mathcal {S}\), see Fig. 3, the volume of resin drained can be computed, \(\Delta V_r^\ell \). Note that knowing the loss of resin it is trivial to compute the current fiber volume fraction.
 9.
On the other hand, from the pressure field it is possible to obtain the vertical component of the reaction force to be applied by the press so as to maintain the actual closure rate. The press force can be computed by integrating the pressure field on the upper boundary \(\mathcal {S}^+\), see Fig. 3. However, note that this force only represents the fluid part, i.e. the resin, but it does not take into account the fiber contribution. In order to account for this extra contribution, we consider the Kim’s model.
 10.
With the press force, \(F^\ell (t_j)\) at hand, the press control can be checked. A NewtonRaphson algorithm is used in order to compute both closure rate and fiber volume fraction correction, denoted by \(\Delta \dot{U}\) and \(\Delta V_f\).
 11.
From the corrections, both closure rate and fiber volume fraction can be updated. At convergence, we set: \(\dot{U}(t_j) \leftarrow \dot{U}^{\ell ^*}(t_j)\) and \(V_f(t_j) \leftarrow V_f^{\ell ^*}(t_j)\), assuming the algorithm converges after \(\ell ^*\) iterations.
 12.
Observe that the semiimplicit linearization allows excluding the thermokinetic module from the nonlinear problem. Otherwise, at each nonlinear iteration, it would be required to come back to the thermokinetic module, to recompute the specific heat, to solve the thermokinetic ROM, and so on.
A Simulation App for the OGV manufacturing process
The Simulation App for the OGV manufacturing process is a processspecific application that allows the user to simulate almost in real time different process conditions and visualize the simulation results. Here we understand by real time simulation the one able to provide the results with no perceptible delay after the user makes its request, i.e. in the order of half a second. However, since the Simulation App is designed to be used directly in the manufacturing facility, the input and output delays are also relevant. As it will be explained later, both input and output interfaces were designed so as to enhance the user’s reactivity by restricting the input data to the bare minimum, while only relevant output information is displayed. Typically, a standard user takes no more than one minute in order to set up a new simulation, while the time consumed in visualizing and interpreting the results depends mostly on the user’s knowledge and experience.
The user interacts with the Simulation App through a basic graphics user interface (GUI). In order to demonstrate the feasibility and potentiality of this kind of applications, we developed a demonstration version in the MATLAB\({\textregistered }\) environment using GUIDE under MacOS X 10.10.5. This allows creating the GUI very easily. The MATLAB Application Compiler\(^{\mathrm{TM}}\) was used in order to create a standalone executable file that could run outside MATLAB\({\textregistered }\) and be easily transferred to the final user for demonstration purposes. The standalone version of the Simulation App has been tested on Microsoft Windows\({\textregistered }\) 7 OS. Other proper implementations are of course possible although not explored in this paper, as they are outside of its scope.
The concept of a Simulation App offers several potentialities. Since the application is processspecific, rather than implementing a general purpose visualisation environment, it is possible to first identify the set of quantities of interest as output and then implement a simple and specific visualization interface. For example, if we know in advance that the maximum pressure gradient is an important indicator for defects, as it is the case in OGV manufacturing processes, we can include a functionality that displays the maximum pressure gradient and its location, at each time step. Then, the user can access to this information by simply activating a checkbox.
 1.
Preprocessing This module performs two basic operations: data loading and parameters conversion. Data loading simply reads all precomputed data required to run the reduced model. This operation only needs to be done once after launching the application, and it is performed in the GUI via the Load Data button. Parameters conversion gathers data entered by the user and computes the simulation parameters. This operation is performed in the GUI via two different tabs: the Parameters Tab, in which some default values that normally do not change are proposed (e.g. resin and fiber specific weight), and the Data Tab, in which process parameters are defined (e.g. temperature cycle, closure rate, etc.) (see Fig. 12). The conversion operation is launched by the Update button, and the initial degree of curing and the initial fiber volume content, both being simulation parameters, are computed.
 2.
Simulation This module is driven by a principal function that governs the interaction between the two reduced models explained in previous sections: POD reduced model for the thermokinetic simulation and PGD reduced model for the consolidation simulation. Basically, this module takes all data defined in the preprocessing module and runs the ROM model in the time interval of interest defined by the user, as explained in “Online simulation strategy: simulation modules coupling” section. The user can also choose the number of equally spaced time frames at which the solution wants to be accessed. Additionally, the user can also demand to access to the solution at particular time frames of interest, such as minimum viscosity time frame, see Fig. 12 for details.
 3.
Visualization This module allows accessing to both field data and quantities of interest, at any desired time frame. This operation is performed by the GUI via two different tabs: the Display Tab and the Quantities of Interest Tab. In the Display Tab, seven scalar fields can be visualized: curing degree, temperature, viscosity, pressure and the three components of the pressure gradient. They are simultaneously displayed on the external boundary of the part as well as in five sections, see Fig. 13. The section view is necessary to appreciate, for instance, the temperature evolution inside the part, which may be higher than in the external boundary due to the exothermic kinetics. Additionally, the location and magnitude of the maximum and minimum of each field can be visualized. In the Quantities of Interest Tab, the time evolution of nine process indicators can be visualized: closure rate, press force, fiber volume fraction or part mass, for instance. See Fig. 14 for details. These are the real meaningful information upon which a process designer can evaluate if the process setting is operating as desired or not.
The RAM consumption of the MATLAB\({\textregistered }\) implementation of the Simulation App was measured. Opening the application requires 239 Mb, whereas the amount of memory increases up to 504 Mb after loading data. The amount of memory required for running the simulation depends on the simulation timespan as well as on the process parameters. In the conditions already described in “Implementation details and results” section, 236 Mb are required on average, which brings the total to 740 Mb. However, the most demanding operation in terms of memory consumption is the Visualization module, which requires up to 1160 Mb. In general, the memory consumption could be drastically reduced in a proper (non demonstrator) implementation of the application. Some parts of the code could also be optimized, for instance, by avoiding reconstruction of the entire fields, since only some slices and the external surface are visualized.
ROMbased Simulation Apps can be a powerful tool for complex composite processes to increase the entitlement yield by adapting for the variation that comes from material chemistry and physical properties in addition to thermal and pressure histories applied during the process. Typical entitlement yield is limited because of the inherent variations and multiphysics interactions. Part quality loss is due to either (i) internal defects, (ii) not meeting dimensional requirements or (iii) poor internal fiber matrix structure. A successful manufacturing process must minimize all three quality components. The Simulation App can be implemented in the manufacturing process seamlessly to make realtime decisions regarding process adaptation possible overcoming inherent variation and truly increasing the entitlement yield of the process. As an example, the Simulation App will take real process measurements as inputs which capture the incoming variation in the preprocessing section and provide quantities of interest e.g. maximum pressure gradient that can be relate to quality. If the quality does not meet requirements, precomputed sensitivity results can be used to identify corrective process change to bring it within requirements. The corrective process change can be as simple as choosing among predefined process cycles. All this can be made possible only because of realtime models capable of reflecting physics of the entire process.
Conclusions
In this paper we have introduced the Simulation App concept, a processspecific simulation tool based on reduced order modeling techniques. Using a combination of them, we were able to reduce from several hours to few seconds the computational time of a coupled multiphysics and strongly nonlinear model describing the manufacturing process of a composite outlet guide vane. Defining a coupling strategy between the different reduced order models was an essential part of this work. In particular, we presented an approach based on an appropriate parametrization of the coupling fields. The use of such fast simulation in a realtime decision making environment being possible, processspecific preprocessing and visualization functionalities were added, leading to the Simulation App concept.
It has also been demonstrated that the Simulation App provides several advantages over generalpurpose simulation software, especially if simulation wants to be used directly in the manufacturing facility. In addition to the computational time reduction, the process specificity of the Simulation App makes it possible to conceive simple yet functional graphic interfaces, for both data input and visualization. The process designer is asked to enter only process parameters while the simulation parameters are limited to the bare minimum. Similarly, the visualization module was designed so as to display only the relevant information, mainly the process indicators upon which decisions are made. Note that, by definition, process indicators are specific to a particular process, which is contrary to the spirit of generalpurpose software. Therefore, the Simulation App establishes a link between process parameters and process indicators through a comprehensive numerical simulation which includes not only the physics and their couplings but also technological constraints such as the control loop of the process, if any.
Finally, even if the Simulation App has been presented on a case study of industrial interest, it is easy to imagine building similar tools for other manufacturing processes involving different physics, materials and technology.
Declarations
Author's contributions
JVA and DB constructed the thermokinetic reduced order model and the Simulation App whereas CG constructed the parametricPGD consolidation model. FL, RU and CB defined the problem, the thermomechanical model, performed material characterization and defined process and control parameters. Finally, FC coordinated and designed the global numerical simulation strategy in collaboration with JVA and DB. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 White SR, Hahn HT. Cure cycle optimization for the reduction of processinginduced residual stresses in composite materials. J Compos Mater. 1993;27(14):1352–78.View ArticleGoogle Scholar
 Pillai VK, Beris AN, Dhurjati P. Intelligent curing of thick section composites using a knowledgebased system. J Compos Mater. 1997;31(1):2251.View ArticleGoogle Scholar
 Shojaei A, Ghaffarian SR, Karimian SMH. Modeling and simulation approaches in the resin transfer molding process: a review. Polymer Compos. 2003;24(4):525–44.View ArticleGoogle Scholar
 Park CH, Woo L. Modeling void formation and unsaturated flow in liquid composite molding processes: a survey and review. J Reinf Plast Compos. 2011;30(11):957–77.View ArticleGoogle Scholar
 Kardos JL, Dudukovic MP, Dave R. Void growth and resin transport during processing of thermosettingmatrix composites. In: Epoxy resins and composites IV. Berlin: Springer, 1986. p. 10123.
 Zobeiry N, Vaziri R, Poursartip A. Computationally efficient pseudoviscoelastic models for evaluation of residual stresses in thermoset polymer composites during cure. Compos Part A Appl Sci Manuf. 2010;41(2):247–56.View ArticleGoogle Scholar
 Mounier AL, Binetruy C, Krawczak P. Multipurpose carbon fiber sensor design for analysis and monitoring of the resin transfer molding of polymer composites. Polymer Compos. 2005;26(5):717730.Google Scholar
 Schmachtenberg E, zur Heide Schulte J, Topker J. Application of ultrasonics for the process control of resin transfer moulding (RTM). Polymer Test. 2005;24(3):330–8.View ArticleGoogle Scholar
 Lawrence JM, Hsiao KT, Don RC, Simacek P, Estrada G, Sozer EM, Stadtfeld HC, Advani SG. An approach to couple mold design and online control to manufacture complex composite parts by resin transfer molding. Compos Part A Appl Sci Manuf. 2002;33(7):981990.View ArticleGoogle Scholar
 Chinesta F, Keunings R, Leygue A. The proper generalized decomposition for advanced numerical simulations: a primer. Berlin: Springer; 2014.View ArticleMATHGoogle Scholar
 Antoulas A, Sorensen D, Gugercin S. A survey of model reduction methods for largescale systems. Contemp Math. 2001;280:193220.MathSciNetMATHGoogle Scholar
 Quarteroni A, Manzoni A, Negri F. Reduced basis methods for partial differential equations: an introduction. New York: Springer International Publishing; 2015.MATHGoogle Scholar
 Volkwein S. Model reduction using proper orthogonal decomposition. Lecture Notes. Graz: Institute of Mathematics and Scientific Computing, University of Graz; 2011. http://www.unigraz.at/imawww/volkwein/POD
 Willcox K, Peraire J. Balanced model reduction via the proper orthogonal decomposition. AIAA J. 2002;40(11):2323–30.View ArticleGoogle Scholar
 Chaturantabut S, Sorensen DC. Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput. 2010;32:2737–64.MathSciNetView ArticleMATHGoogle Scholar
 Peherstorfer B, Butnaru D, Willcox K, Bungartz HJ. Localized discrete empirical interpolation method. SIAM J Sci Comput. 2014;36(1):A168–92.MathSciNetView ArticleMATHGoogle Scholar
 Willcox K. Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition. Comput Fluids. 2006;35(2):208226.View ArticleMATHGoogle Scholar
 Ryckelynck D, Vincent F, Cantournet S. Multidimensional a priori hyperreduction of mechanical models involving internal variables. Comput Methods Appl Mech Eng. 2012;225:28–43.MathSciNetView ArticleMATHGoogle Scholar
 Amsallem D, Zahr MJ, Choi Y, Farhat C. Design optimization using hyperreducedorder models. Struct Multidiscip Optim. 2015;51(4):919–40.MathSciNetView ArticleGoogle Scholar
 Benner P, Gugercin S, Willcox K. A survey of projectionbased model reduction methods for parametric dynamical systems. SIAM Rev. 2015;57(4):483–531.MathSciNetView ArticleMATHGoogle Scholar
 Ryckelynck D, Chinesta F, Cueto E, Ammar A. On the a priori model reduction: overview and recent developments. Arch Comput Methods Eng State Art Rev. 2006;13(1):91–128.MathSciNetView ArticleMATHGoogle Scholar
 Patera AT, Rozza G. Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Copyright MIT (2006–2015). MIT Pappalardo Monographs in Mechanical Engineering. 2007. http://augustine.mit.edu.
 Drohmann M, Haasdonk B, Ohlberger M. Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J Sci Comput. 2012;34(2):A937–69.MathSciNetView ArticleMATHGoogle Scholar
 Fritzen F, Leuschner M. Reduced basis hybrid computational homogenization based on a mixed incremental formulation. Comput Methods Appl Mech Eng. 2013;260:143–54.MathSciNetView ArticleMATHGoogle Scholar
 Rozza G, Huynh DBP, Patera AT. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Archiv Comput Methods Eng. 2008;15(3):229–75.View ArticleMATHGoogle Scholar
 Rozza G. Fundamentals of reduced basis method for problems governed by parametrized PDEs and applications. In: Ladeveze P, Chinesta F, editors. CISM Lectures notes. Separated Representation and PGD based model reduction: fundamentals and applications. Vienna: Springer; 2014.
 Hesthaven J, Rozza G, Stamm B. Certified reduced basis methods for parametrized partial differential equations. Berlin: Springer; 2015.MATHGoogle Scholar
 Ammar A, Mokdad B, Chinesta F, Keunings R. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J Non Newton Fluid Mech. 2006;139:153–76.View ArticleMATHGoogle Scholar
 Ammar A, Mokdad B, Chinesta F, Keunings R. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Part II: transient simulation using spacetime separated representation. J Non Newton Fluid Mech. 2007;144:98–121.View ArticleMATHGoogle Scholar
 Falco A, Nouy A. A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional EckartYoung approach. J Math Anal Appl. 2011;376:469480.MathSciNetView ArticleMATHGoogle Scholar
 Ladevèze P. The large time increment method for the analyze of structures with nonlinear constitutive relation described by internal variables. Comptes Rendus Académie des Sciences Paris. 1989;309:1095–9.MATHGoogle Scholar
 Boucinha L, Gravouil A, Ammar A. Spacetime proper generalized decompositions for the resolution of transient elastodynamic models. Comput Methods Appl Mech Eng. 2014;255:67–88.MathSciNetView ArticleMATHGoogle Scholar
 Chinesta F, Ammar A, Leygue A, Keunings R. An overview of the Proper Generalized Decomposition with applications in computational rheology. J Non Newton Fluid Mech. 2011;166:578–92.View ArticleMATHGoogle Scholar
 Hackbusch W. Tensor spaces and numerical tensor calculus. 1st ed. BerlinHeidelberg: Springer; 2012.View ArticleMATHGoogle Scholar
 Lavedèze P, Chamoin L. On the verification of model reduction methods based on the Proper Generalized Decomposition. Comput Methods Appl Mech Eng. 2011;200(23–24):2032–47.MathSciNetMATHGoogle Scholar
 Nouy A. A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Comput Methods Appl Mech Eng. 2007;196:4521–37.MathSciNetView ArticleMATHGoogle Scholar
 Bognet B, Leygue A, Chinesta F. Separated representations of 3D elastic solutions in shell geometries. Adv Modell Simul Eng Sci. 2014;1(1):1–4.MATHGoogle Scholar
 Chinesta F, Leygue A, Bordeu F, Aguado JV, Cueto E, Gonzalez D, Alfaro I, Ammar A, Huerta A. PGD based computational vademecum for efficient design, optimization and control. Arch Comput Methods Eng. 2013;20(1):31–59.MathSciNetView ArticleGoogle Scholar
 Chinesta F, Leygue A, Bognet B, Ghnatios Ch, Poulhaon F, Bordeu F, Barasinski A, Poitou A, Chatel S, MaisonLePoec S. First steps towards an advanced simulation of composites manufacturing by automated tape placement. Int J Mater Form. 2014;7(1):81–92.View ArticleGoogle Scholar
 Borzacchiello D, Aguado JV, Chinesta F. Reduced order modelling for efficient process optimisation of a hotwall chemical vapour deposition reactor. Int J Numer Methods Heat Fluid Flow. In press.
 Kamal MR, Sourour S. Kinetics and thermal characterization of thermoset cure. Polymer Eng Sci. 1973;13(1):59–64.View ArticleGoogle Scholar
 Gebart B. Permeability of unidirectional reinforcements for RTM. J Compos Mater. 1992;26(8):1100–33.View ArticleGoogle Scholar
 Barrault M, Maday Y, Nguyen NC, Patera AT. An “empirical interpolation” method: application to efficient reducedbasis discretization of partial differential equations. Comptes Rendus Mathématique. 2004;339(9):667–72.MathSciNetView ArticleMATHGoogle Scholar
 Germoso C, Aguado JV, Fraile A, Alarcón E, Chinesta F. Efficient PGDbased dynamic calculation of nonlinear soil behavior. Comptes Rendus Mecanique. 2016;344(1):24–41.View ArticleGoogle Scholar