The heat transfer problem is solved in the domain representing a composite laminate blank. It is considered to be an arbitrary curved thin shell, where the local positions are located via a curvilinear parallel coordinate system. A local frame can be attached to each point. Coordinate enables the location of points along the thickness direction, that is to say along, the normal vector to the shell mid-plane (see Figure 1). In this domain the compo- site material is considered to be a continuous medium with effective homogene- ous properties.

In the considered heat transfer problem, the conduction is assumed to be governed by an anisotropic Fourier law where the local heat flux is written as:

where is the thermal conductivity tensor, the temperature field and the spatial derivative operator. In the present work, it is assumed that the through thickness direction is a principal direction of the thermal conduc- tivity. This is a classical assumption in the case of standard composite laminates [10] [25] . Thus, in the basis, it writes

being the in-plane thermal conductivity tensor and the through thickness thermal conductivity. Note that this hypothesis fails in the case of complex 3D architectured composites. Defining the in-plane surface gradient, Equation (2) can be separated into a through thickness and an in-plane fluxes:

In the case of a flat shell, the coordinate system is the natural cartesian coordinate system, and. In the more complex case of an arbitrary curved shell, the reader should refer to Appendix for a proper definition of the surface gradient. This demons- tration shows that in the case of a thin shell with small curvature, the operator does not depend on the through thickness position.

Using this separation, without internal heat source in the domain, the energy balance typically writes

being the density of the composite material and its specific heat. Once again, for a flat shell, the surface divergence, but for curved shell, it is defined in Appendix and it is constant through thickness.

The domain is bounded by the boundaries, and, as defined in Figure 1. For the sake of simplicity, the lateral boundaries are considered insulated:

being the outward normal to each surface. Conversely, in order to accura- tely model temperature history imposed on the upper and lower boundaries and, a mixed boundary condition is assumed:

where (respectively) is the temperature imposed on the upper (re- spectively lower) boundary and (respectively) is the heat exchange coefficient. This mixed boundary condition modelling can account for non ideal contact with the mould [11] [12] . In its limit form, it is also suited to model both Dirichlet or Neumann boundary conditions. Note that the development pro- posed hereunder could seemlessly be conducted with any type of boundary conditions (temperature imposed, heat flux, radiating surface...).

The initial temperature field, assumed given, is defined as:

The first step in the proposed model reduction is to seek the solution of the system of Equations (5) to (8) as a sum of a through thickness averaged field and of a fluctuation field:

where the operator

is the through thickness average, being the local shell thickness. It is obvious that using this additive decomposition, the average field does not depend on the -coordinate whereas the fluctuation field has a zero thickness average. This decomposition is intuitive and does not necessitate any assump- tion. Substituting this decomposition (9) in the heat Equation (5), considering constant material properties, and noting that and the operator do not depend on the -coordinate gives:

Applying the average operator on both hands of this equation leads to

By defining the upper and lower inward boundary fluxes

Equation (12) writes:

which is the average field heat equation. It rules the in-plane mean field tem- perature evolution. Subtracting this mean heat equation from Equation (11) results in the fluctuating heat equation:

which rules the through thickness temperature fluctuation.

Assuming a thin plate for which, the so called aspect ratio for conduc- tion:

and the dimensional analysis safely leads to

Equation (15) then reduces to the fluctuating field heat equation:

Equations (14) and (18) achieve a decomposition of the initial heat Equation (5) in the average and fluctuating contributions. Nonetheless, without further assumptions, these two equations are strongly coupled through the source terms.

Reduced model. Summing Equations (14) and (18), and adding the term, gives

This equation, along with boundary and initial conditions (6), (7) and (8) defines the reduced boundary value problem (). In the bulk Equation (19), the first spatial differential operator of the right hand side acts on the average parts of the temperature field only. The solving of this reduced boundary value problem is therefore not straightforward. In the next section, a numerical method is proposed to solve this original model. It will also confirm its well-posedness.

Time discretization. The time evolution problem given by Equation (19) is solved in the framework of a standard incremental iterative time integration scheme. At a given time, the solution is supposed to be known. Then, the solution of the reduced boundary value problem defined above is searched at next time step.

Any conventional time integration scheme, such as for example explicit or implicit schemes, can be used to determined in terms of, so that the developments detailed hereunder will easily be implemented in such software environment.

Operator splitting. To solve Equation (19), an operator splitting method is used. This numerical method enables to solve evolution equations that involve a sum of differential operators (see for example [20] ). Adopting the splitting ini- tially suggested by Douglas [18] and later called locally one-dimensional (LOD) method (see for instance [26] and references therein), the two differential opera- tors in the right hand side of Equation (19) are considered separately. Note that in this linear case, the proposed splitting does not introduce additional nu- merical error beside the time integration error [20] . As illustrated in Figure 2, a so-called fractional time step method is adopted, where two problems are solved successively, each one containing one of the operators:

• Step 1: solve the following 1D boundary value problem called () over one full time step

gives the intermediate result at the end of the time step.

• Step 2: solve the 2D boundary value problem over one full time step

where the initial condition is the value of the field computed in step 1. The solution of this second step at the end of the time step () is identified to.

Whereas the system () defined in step 1 is a well posed unidimensional partial differential equation, it is somewhat disturbing that both and appear in the problem (21) defined in step 2.

ADI model. To ensure the well-posedness of this step 2, the additive decom- position (9) is again substituted in system (21). Applying the average operator gives the in-plane boundary value problem ()

Finally, subtracting (22) from (21) results in

which admits the trivial constant solution:

Therefore, the fluctuating part of this second step is simply the fluctuating part of computed in the first step. In other terms, this second step does not introduce any additional out-of-plane fluctuation to the solution.

A square flat plate of dimensions and thickness is considered. The origin of the cartesian coordinate system is taken in the centre of the plate. In such a flat plate case, the curvilinear coordinates are identified to the cartesian ones: and.

Material properties. In this test case, a PA66/glass fibre composite material is considered. The homogenized material properties are adapted from the litera- ture [27] . The in plane conductivity is considered isotropic and all the material properties are supposed constant and are listed in Table 1.

Boundary and initial conditions. The boundary and initial conditions are given in Table 2. The plate is supposed to be initially at uniform room tempera- ture.

A different heating condition is imposed on the upper and lower surfaces with. It is representative of the temperature imposed by a hot mould con- tact. In order to add in-plane variability to the problem, the exchange coeffi- cients and artificially depend on space position through the characteristic gaussian function

The problem is solved on the time interval.

Numerical methods. The 1D transient boundary value problems () and the 2D transient boundary value problem () are solved using a finite element method with piecewise linear interpolation. An implicit (backward Euler) time integration scheme is used for all time integrations. The proposed algorithm was programmed in MATLAB^®, which enables the parallel resolution of the () problems.

Mesh. For the reference simulation, a 3D regular mesh made of 3600 hexa- hedron is obtained by extruding a regular in-plane 2D mesh that consists of quadrangular elements. There are thus 30 elements in the thickness, and in terms of nodes, and.

For the proposed separated method, the mesh consists of the same 31 nodes through the thickness for the problems and of a triangular regular mesh with the same 3721 nodes for the problem.

The interpolations used in every finite element methods (3D in COMSOL, 2D in and 1D in) are all linear, which enables to expect for the same precision.

Time step. Time stepping in the FEM reference simulation follows the COMSOL built-in algorithm and is forced not to exceed. The time integration scheme is a standard backward difference scheme. On the contrary, a constant time step is used in the presented method. In this first test case, the time steps for both and problems are the same.

The convergence of the numerical methods used was first validated on a standard one-dimensional test case by comparing the numerical solution with an analytical solution given by Jaeger [28] .

Figure 4 shows the in-plane temperature profiles at three different heights, at final time. Figure 5 represents the through thickness temperature

profiles in the centre and on the edge of the plate at and final time. The figures show a good superposition of the reference field obtained with the finite element simulation and the one obtained with the presented method. The same numerical artifact (a slight oscillation) is found with both methods in the through thickness profile at early time (). This is due to the finite element and time discretization that fail to accurately predict thermal shocks. this artifact does not influence the later time predictions (see for instance Fachinotti and Bellet [29] regarding this issue).

The maximum residual relative error

is defined, where is the field computed with the 3D model in COMSOL and is the field computed with the presented method. At final time, the error does not exceed which represents around.

The reference finite element simulation was computed in on a desktop computer (see Table 3). The solving time per time step was about. The separated form solution was computed on the same computer in no more than, with about per time step. This represents a speed up of over 28

times. Using asynchronous time steps for and results in an additional reduction in the total computational time. Moreover, the problems are solved in a sequential manner in this test case. Solving them in parallel results in additional speed-up.

The limits of the proposed resolution strategy are investigated in this section. It is reminded that two conditions were required in the model development:

1) A small aspect ratio for conduction such that Equation (18) stands. This corresponds to the thin-shell assumption in the case where the in-plane and through thickness conductivities are of the same order of magnitude.

2) In the case of a curved shell domain, the radii of curvature should be large compared to the shell thickness. This ensures that the metrics given in the Appendix do not depend on the coordinate.

Thick part. In the test case presented above, the aspect ratio for conduction is very small () which explains the good app- licability of the thin plate assumption and the presented reduced method. The limit imposed by the first condition above was investigated by performing additional simulations with larger values of. With this aim, the plate dimen- sion was decreased. The plate is still flat and square. As shown in Figure 6 if stays smaller that, the thin plate assumption stands and the error given by (26) between the 3D finite element reference solution and the separated form solution does not exceed. It would even fall to lower than 1% for typical part shape encountered in composites processing ().

Sharp curvature. In order to investigate the curvature limit imposed by the second condition discussed above, a curved shell was considered. The domain is now an L-shape blank of length, with two flanges of length and a radius of curvature of the mid-plane surface. The blank thickness is kept (see Figure 7).

The boundary conditions on the upper and lower surfaces are now such that and. The reference field computed with the full 3D formulation using COMSOL Multiphysics^® and the

field obtained using the proposed strategy are computed for the time range. The through-thickness profile along the first diagonal schematized in Figure 7 is plotted in Figure 8.

As the blank thickness to radius of curvature ratio gets larger, the metric tensor given in Appendix by Equation (30) depends on the through thickness position. Thus Equation (31) does not stand and the proposed decomposition strategy fails at predicting the initial 3D problem. This is the case for Figure 8 where the thickness to radius ratio.

To identify the limit of applicability, several simulations with varying radius of curvature were performed. As shown in Figure 9 if stays below 0.2, which is usually the case in industrial geometries, the error between the 3D finite element reference solution and the separated form is below.

The proposed ADI resolution method was applied to an industrial case representative of the thermostamping process. A thick laminate comprised of 16 anisotropic plies stacked on a sequence is considered. The temperature dependant thermal properties are adapted from carbon fibre reinforced PEEK and are given in Table 4. The initially hot laminate (at) comes in contact with a cold matrix and punch set, as illustrated in Figure 10, at time.

The 2D heat transfer problem is solved using (i) a full 2D resolution in COMSOL (ii) the presented alternate direction implicit (ADI) method, and (iii) a series of independant one-dimensional through thickness problems. In the ADI method, the problem consists of a 1D homogenized through thickness problem. Because of the stacking sequence, the in-plane thermal conductivity tensor is isotropic and is an average of the longitudinal and transsverse properties given in Table 4. Nonlinear resolution is performed in MATLAB over a physical time of with 150 time steps. In COMSOL, the exact multiply description is implemented. Using symmetry, only half of the geometry is considered and presented hereunder.

Three-dimensional effect. The problem is nonlinear, and, as visible in Figure 11, highly three-dimensional at the vicinity of the shear edge (between the

punch and the matrix). Still the proposed ADI method is able to partly discribe this tridimensional effect thanks to the problem that considers in plane diffusion.

Temperature profiles at three different positions at time are plotted in Figure 12.

• Far from the shear edge (cut CC'), the temperature gradient is mostly through thickness and the three approaches prove efficient at describing the through thickness temperature field.

• In the centre of the shear edge zone (cut AA'), the ADI method enables an accurate recovery of the through thickness profile obtained with the full 2D method. On the contrary, at this position AA', the one-dimensional method highly overestimates the temperature since it does not account for the nearby cold moulds.

• Similarly in the intermediate region over the matrix (cut BB’), the one- dimensional approach under predicts the temperature field. On the contrary, the ADI proposed method, enables a partial description of the three-dimensional effects (). Nonetheless, the method results in overpredicting temperature at height. At this worst position, three-dimensional effects are en- hanced, the decomposition methods fails and this artifact (also visible in Figure 11) cannot be overcome.

Nonlinearity. In addition to this three-dimensional effect, the proposed in- dustrial case is nonlinear, since the properties are temperature dependant. In this nonlinear case, the ADI method still proved efficient at predicting the temperature field. The efficiency of the method is explained by the very smooth non-linearities of the thermal properties used in the test case (see Table 4). Given that this is the case for the majority of industrial thermoplastic composite, the decomposition ADI method will likely be efficient for other industrial materials.

Multiply. Finally, the industrial test case was performed with a 16 plies laminates, with a very harsh anisotropic stacking. The ADI, which considers an homogenized in-plane conductivity for the in-plane problem, still proves efficient at predicting the thermal fields. In conclusion, as far as the heat transfer is concerned, a multiply stacking representative of an industrial blank can be considered as homogeneous through thickness.