Composites Part A: Applied Science and Manufacturing
Methodological approach to efficient modeling and optimization of thermal processes taking place in a die: Application to pultrusion
Introduction
The main aim of this work is proposing an original methodological approach to efficient numerical modeling and optimization. More specifically we are focusing on thermal processes taking place in a die, although the methodology proposed in this paper could easily be extended to many other processes.
In general optimization implies the definition of a cost function and the search of the optimum process parameters (e.g. temperatures of the heaters, temperature of the material coming to the die, flow rate, … in the case of materials flowing into a heated die) defining the minimum of that cost function. The process starts by choosing a tentative set of process parameters. Then the process is simulated by discretizing the equations defining the model of the process. The solution of the model is the most costly step of the optimization procedure. As soon as that 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. Obviously, nothing ensures that such minimum is global, so more sophisticated procedures exist in order to explore the domain defined by the parameters and escape from local minimums traps. The parameter updating is carried out in a way ensuring the reduction of the cost function. Many techniques update the model parameters in order to move along the cost function gradient. However, for identifying the direction of the gradient one should compute not only the fields involved in the model but also the derivatives of such fields with respect to the different process parameters. The evaluation of these derivatives is not in general an easy task. Conceptually, one could imagine that by perturbing slightly only one of the parameters involved in the process optimization and then solving the resulting model, one could estimate, using a finite difference formula, the derivative of the cost function with respect to the perturbed parameter. By perturbing sequentially all the parameters we could have access to the derivatives of the cost function with respect to all the process parameters (that is, the sensibilities) that define the cost function gradient, on which the new trial set of parameters should be chosen. There are many strategies for updating the set of process parameters and the interested reader can find most of them in the books focusing on optimization procedures.
Our interest here is not the discussion on particular optimization strategies, but pointing out that standard optimization strategies need numerous direct solutions of the problem that represents the process, one solution for each tentative choice of the process parameters. The solution of such models is a tricky task that demands important computational resources and usually implies extremely large computing times.
In this paper we propose a radically different approach, to the authors’ knowledge never explored. The approach here proposed considers the unknown process parameters as new coordinates of the model. In fact, coordinates, or space dimensions, represent the (non-necessarily physical) locations at which the solution is to be represented. Thus, strictly speaking, one could compute the solution of the problem for any value of the unknown parameters (in a bounded interval). This converts those unknown parameters in new dimensions of the space in which the model is defined. This idea seems exciting but it involves a major difficulty.
This strategy faces a challenging problem if the number of parameters of the model increases. It is well known that the number of degrees of freedom for a mesh-based discretization technique (say, finite element, finite difference, …) increases exponentially with the number of dimensions. Thus, for a hypercubic domain, the number of degrees of freedom scales with the number of nodes along each spatial direction to the power of the number of dimensions. For instance, in 2D and considering 100 nodes along each direction with a single degree of freedom per node, the resulting number of degrees of freedom becomes 1002. In 3D, the number of degrees of freedom rises to 1003 and so on. This exponential increase of the number of degrees of freedom can be literally out of reach for nowadays computers even if the number of dimensions increases only moderately. This phenomenon is known as curse of dimensionality.
Of course, to efficiently deal with this problem a strategy different of mesh-based discretization methods should be employed. Although efficient techniques exist for moderate number of spatial dimensions, such as sparse grid methods [4], they fail when the dimensionality increases. Here, we consider the use of Proper Generalized Decompositions (PGD) [1], [2], [7], [8]. PGD techniques construct an approximation of the solution by means of a sequence of products of separable functions. These functions are determined “on the fly”, as the method proceeds, with no initial assumption on their form.
The PGD method, while it can be considered as a model reduction technique (and hence its name, a generalization of the Proper Orthogonal Decomposition – POD –) can deal very efficiently with highly multi-dimensional problems, since only a sequence of low-dimensional problems is solved. Details of the technique are provided in Section 2 of this paper.
Once an efficient strategy of dealing with high-dimensional solutions has been defined, the numerical solution of problems with unknown data becomes straightforward. As mentioned before, the strategy here proposed consists of considering the unknown parameters as new coordinates of the model. Thus, the solution is computed only once and it allows to have access to the unknown field, as well as to the explicit expression of its derivatives, for any possible choice of the model parameters by a simple particularization of the parametric solution, that is, by a simple postprocessing.
As can be readily noticed, the potential of the technique for inverse identification, optimization, etc. seems to be huge. In this paper we propose a methodological approach in this direction. In what follows we are considering a thermal model of a material moving through a die equipped with some heating devices on the die walls. We could consider as process parameters the temperatures prescribed in the different heaters, the flow rate, the temperature of the material coming into the die, etc. For the sake of simplicity in what follows we are restricting the parametric space to the heating devices temperatures. The choice of the cost function depends on each particular process. There are many choices and because in this work we are more interested in proposing and illustrating a new methodological modeling and optimization approach than in analyzing deeply a particular process, we will restrict our analysis to a simplified model of pultrusion processes. Pultrusion is a continuous process to produce constant cross-sectional profile composites. During this process, fiber reinforcements are saturated with resin, which are then pulled through a heated die. The resin gradually cures inside the die while generating heat. At the exit, pullers draw the composite out and a traveling cut-off saw cuts it at the designed length. We consider the process conditions described in [5] and sketched in Fig. 1.1
For decades, engineers have relied on experience to define optimal parameters for pultrusion processes, pushed by the popularization of this technique in industry [10], [12]. Nowadays, this process has been extended to thermoplastic resins as well as to reactive systems, in which the monomers polymerize inside the die, an efficient route for considering high viscosity thermoplastic resins. In these scenarios new efficient optimization procedures are urgently needed. Standard optimization approaches and others based on the use of genetic algorithms have been recently proposed and applied (see for example [5] and the references therein. However the efficiency of those approaches seems to be limited to a reduced number of process parameters because one must solve a thermal model for each choice of the process parameters, and it is well known that when the dimension of the parametric space increases the exploration of the space defined by the process parameters becomes more and more arduous, needing for numerous, sometimes excessive, solutions of the model governing the process.
If we consider the thermal model related to the pultrusion process sketched in Fig. 1, whose parametric space reduces to the temperatures prescribed at the three heating devices, θ1, θ2 and θ3, we could summarize traditional optimization procedures as follows:
- •
Until reaching a minimum of the cost function proceed by:
- 1.
Computing the temperature field related to the trial choice of the process parameters, i.e. u(x;θ1,θ2,θ3).
- 2.
Computing the cost function from the just calculated thermal field.
- 3.
Checking the optimality. While the optimum is not reached, update the process parameters by using an appropriate strategy and comeback to step 1 for another solution of the thermal model for the process parameters just updated.
- 1.
In the approach that we propose in this work the procedure is substantially different. It proceeds as follows:
- •
Compute the thermal field for any possible choice of the process parameters: u(x, θ1, θ2, θ3) (here the heaters temperatures play the same role that the space coordinates), the problem becoming multi-dimensional.
- •
Until reaching a minimum of the cost function proceed by:
- 1.
Particularizing the parametric solution to the considered values of the process parameters.
- 2.
Computing the cost function from the just calculated thermal field.
- 3.
Checking the optimality. While the optimum is not reached, update the process parameters by using an appropriate strategy and comeback to step 1 for another particularization of the parametric solution.
- 1.
Thus, in our proposal the thermal model is solved only once and then it is particularized for any choice of the process parameters. The price to pay is the necessity of solving a multi-dimensional thermal model that now has as coordinates the physical space x and all the process parameters, i.e. the three heaters temperatures in the example addressed here.
Obviously, the solution of the resulting multi-dimensional model is a tricky task if one consider a standard mesh based discretization strategy because the number of degrees of freedom increases exponentially with the dimensionality of the model. To circumvent this serious difficulty, also known as curse of dimensionality, we consider a separated representation of the temperature field in the PGD framework, in which the temperature reads:To build up such separated representation we only need to compute the functions defined in the space domain Fi(x) and the one-dimensional functions Θji(θj), j = 1,2,3, defined in the intervals in which the heaters temperatures can evolve.
In Section 2 we revisit the main ideas involved in the construction of such separated representation. In Section 3 we describe the procedure to include the boundary conditions as extra-coordinates. In Section 4 a simplified pultrusion model is addressed. Then, in Section 5 we consider one possible optimization strategy based on the combination of the already computed parametric solution with a minimization strategy based on the use of moving least squares on the response surface. Due to the methodological purposes of this paper a simple cost function will be considered, enforcing a constant temperature of the flowing material on die outlet. Despite the simple scenario here addressed the reader can appreciate the huge potentialities that such approach represents.
Section snippets
Illustrating the solution of multi-dimensional parametric models by using the PGD
Imagine for example that you are interested in solving the heat equation but that you do not know the material thermal conductivity, because it has a stochastic nature or simply because prior to solve the thermal model you should measure it. You have three possibilities: (i) you wait to know the conductivity before solving the heat equation (a conservative solution!); (ii) you solve the equation for many values of the conductivity (a sort of Monte Carlo) and then the work is done (a sort of
Parametric boundary conditions
Very often, in the optimization of an industrial process, it is necessary to solve the problem for different boundary conditions. Boundary conditions do not behave as any other parameter in the PGD, and therefore deserve some additional comments. In general, it is needed to perform a change of variable to introduce the boundary condition into the differential equation and then define it as an extra coordinate. To illustrate this procedure we consider, for the sake of simplicity and without any
Parametric thermal model of a heated die in pultrusion processes
In modeling the pultrusion process, as sketched in Fig. 1, we consider the thermal process within the die as modeled by the following two-dimensional convection–diffusion equation:where k is the thermal conductivity, q is the internal heat generated by the resin curing reaction, ρ is the density, C is the specific heat and v is the extruded profile speed. The material flowing inside the die is in contact with the die wall. Thus, conduction is the only heat transfer mechanism
Optimization strategy
In this section we consider the parametric solution already computed:
The objective of the optimization procedure consists of the determination of the process parameters θ1, θ2 and θ3 in order to minimize an appropriate cost function depending on the considered physics. In this work we are only interested in describing the main ingredients involved in the proposed optimization strategy, and for this reason, in what follows, we consider a cost
Conclusion
In this paper we have presented the abilities of the Proper Generalized Decomposition for solving a multi-dimensional problem in which parameters involved in optimization procedures are introduced as extra-coordinates in the model. Using the PGD the solution complexity scales linearly with the dimension of the space. The method seems to be able to deal with optimization problems whose deterministic solutions were a dream until now.
Advanced modeling of thermal processes taking place in a heated
Acknowledgement
This work has been partially supported by the Spanish Ministry of Science and Innovation, through Grant No. CICYT-DPI2011–27778-C02–01.
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