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Thermostamping of thermoplastic matrix composites is a process where a preheated blank is rapidly shaped in a cold matching mould. Predictive modelling of the main physical phenomena occurring in this process requires an accurate prediction of the temperature field. In this paper, a numerical method is proposed to simulate this heat transfer. The initial three-dimensional heat equation is handled using an additive decomposition, a thin shell assumption, and an operator splitting strategy. An adapted resolution algorithm is then presented. It results in an alternate direction implicit decomposition: the problem is solved successively as a 2D surface problem and several one-dimensional through thickness problems. The strategy was fully validated versus a 3D calculation on a simple test case and the proposed strategy is shown to enable a tremendous calculation speed up. The limits of applicability of this method are investigated with two parametric studies, one on the thickness to width ratio and the other one on the effect of curvature. These conditions are usually fulfilled in industrial cases. Finally, even though the method was developed under linear assumption (constant material properties), the strategy validity is extended to multiply, temperature dependant (nonlinear) case using an industrial test case. Because of the standard methods involved, the proposed ADI method can readily be implemented in existing software.

Thermoplastic composites offer new possibilities for the industry. Large struc- tures can be processed rapidly and more cost-effectively than when thermoset composites are used, since the latter need to undergo lengthy curing reactions. The ability to fuse thermoplastic resins gives new perspectives for forming processes.

The thermostamping process is derived from the metallic materials industry. Forming occurs in two steps. In a first step, a semi-finished thermoplastic flat laminate, called the blank, is heated above the processing temperature of the matrix, usually using infra-red lamps. In the second step, this hot blank is quickly transferred to a cooled mould where it is stamped and given its final shape [

Even though metal stamping has been the subject of extensive research work in the past decades (see for instance the review by Karbasian and Tekkaya [

It is well established that the temperature evolution is of major importance in this forming process. Keeping this in mind, de Luca et al. [

A finer through thickness temperature distribution description was proposed by Thomann et al. [

Furthermore, the proposed model is designed to be easily implemented in any existing industrial code (such as Plasfib [

Considering the composite blank as a thin shell, it is natural to decompose the 3D temperature solution into a shape function and an in-plane temperature. As suggested by Saetta and Rega [

With this decomposition, the accuracy of through thickness description de- pends on the type of shape functions chosen. Within this framework, some authors suggested to construct new 3D shell finite elements that integrate this through thickness heat transfer effects [

Adopting a fine through-thickness discretization therefore seems a more flexible approach, though potentially time-consuming. In this idea, Bognet et al. [

where the shape functions, themselves, are described with a fine discretization involving hundreds of degrees of freedom. In this framework, Bognet et al. considered a series of multiplicative shape functions, where each mode

In this paper, starting from a very general approximation framework as given by Equation (1), we propose a reduced numerical scheme, adapted to thin compo- site shells, that preserves the three-dimensional nature of the heat transfer problem but takes advantage of the good physical separation between in-plane and out-of-plane phenomena, even in case of anisotropic thermal properties. The present method is based on an operator splitting technique that enables to simplify a time evolution problem implying several spatial dimensions. The general framework of operator splitting techniques always considers an incre- mental iterative time integration strategy. Over 50 years ago, Douglas [

Following these ideas, the present paper proposes an operator splitting strategy adapted to the composite shell problems to solve the reduced heat trans- fer model. In fine, this results in two separated problems. A solving algorithm and numerical implementation is then proposed. The approach is validated on a flat plate test case, and its limits are determined with parametric studies. The method validity is extended to nonlinear cases with an industrial appli- cation.

The heat transfer problem is solved in the domain

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

where

In the case of a flat shell

Using this separation, without internal heat source in the domain

The domain

where

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

This section presents a reduction of the heat transfer problem defined above. The reduced boundary value problem is obtained thanks to an intuitive decom- position of the temperature field and a thin shell assumption. An implementa- tion strategy is then proposed to numerically solve this problem. Here, for the sake of clarity, the heat transfer problem is assumed linear (the material pro- perties

The first step in the proposed model reduction is to seek the solution

where the operator

is the through thickness average,

Applying the average operator

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

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

This equation, along with boundary and initial conditions (6), (7) and (8) defines the reduced boundary value problem (

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

Any conventional time integration scheme, such as for example explicit or implicit schemes, can be used to determined

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 [

• Step 1: solve the following 1D boundary value problem called (

gives the intermediate result

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

where the initial condition

Whereas the system (

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

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

which admits the trivial constant solution:

Therefore, the fluctuating part

To ensure spatial numerical integration of this problems, a spatial discretization has to be adopted. Within the defined shell like domain

Resolution scheme.

Following the above additive decomposition and operator splitting strategy,

In this sum,

•

•

•

Expected computational speed up. A conventional in plane discretization of an industrial geometry would typically result in

Additionally, because of the high through thickness temperature gradients associated with thermal shocks that occur in thermo-stamping, a fine through thickness discretization is required, for instance

Solving the initial 3D heat transfer problem defined in Section 2.1 using standard methods would result in solving a transient problem with

On the contrary, in the proposed resolution strategy, at each time step,

Asynchronous time integration. Because of the thin plate assumption where

In practice, the global resolution algorithm presented in

In this section, first, the proposed separated model and resolution strategy is validated on a test case that largely fulfills the thin shell assumption. Then the speed up is discussed and the limits of the presented model are investigated with rougher cases (thick and curved shell).

In order to validate the proposed resolution strategy, the temperature fields obtained using the presented model are compared with the temperature fields obtained by solving the initial three-dimensional problem, using a commercial software (COMSOL Multiphysics 5.0^{®}).

A square flat plate of dimensions

Material properties. In this test case, a PA66/glass fibre composite material is considered. The homogenized material properties are adapted from the litera- ture [

Boundary and initial conditions. The boundary and initial conditions are given in

A different heating condition is imposed on the upper and lower surfaces with

The problem is solved on the time interval

Numerical methods. The 1D transient boundary value problems (^{®}, which enables the parallel resolution of the (

Density | ||
---|---|---|

Specific heat | ||

In plane conductivity | ||

Out of plane conductivity |

Initial temperature | ||
---|---|---|

Exchange coefficients | ||

Imposed temperature | ||

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

For the proposed separated method, the mesh consists of the same 31 nodes through the thickness for the

The interpolations used in every finite element methods (3D in COMSOL, 2D in

Time step. Time stepping in the FEM reference simulation follows the COMSOL built-in algorithm and is forced not to exceed

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 [

profiles in the centre and on the edge of the plate at

The maximum residual relative error

is defined, where

The reference finite element simulation was computed in

Test Case | CPU time per time step | CPU time |
---|---|---|

COMSOL 3D | 5 s | 10,000 s |

Proposed method, synchronous | 0.178 s | 356 s |

0.022 s | ||

0.145 s | ||

Proposed method, asynchronous | 300 s |

times. Using asynchronous time steps for

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

2) In the case of a curved shell domain

Thick part. In the test case presented above, the aspect ratio for conduction

Sharp curvature. In order to investigate the curvature limit imposed by the second condition discussed above, a curved shell was considered. The domain

The boundary conditions on the upper and lower surfaces are now such that ^{®} and the

field

As the blank thickness to radius of curvature

To identify the limit of applicability, several simulations with varying radius of curvature

The proposed ADI resolution method was applied to an industrial case representative of the thermostamping process. A

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

Three-dimensional effect. The problem is nonlinear, and, as visible in

Transverse thermal conductivity | ||
---|---|---|

Longitudinal thermal conductivity | ||

Specific heat | ||

Density |

punch and the matrix). Still the proposed ADI method is able to partly discribe this tridimensional effect thanks to the

Temperature profiles at three different positions at time

• 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 (

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

Multiply. Finally, the industrial test case was performed with a 16 plies laminates, with a very harsh

Several thermostamping simulation tools exist which handle the mechanics. This is the case, for instance, of Plasfib [

1) In these tools, the global time integration scheme is incremental and therefore follows the same scheme as the one described in Section 2.2.2. The iterative time integration procedure is thus consistent between the existing mechanical algorithm, and the proposed heat transfer with operator splitting algorithm.

2) The two-dimensional problem

3) The problems

4) The through thickness average two-dimensional temperature field

An alternate direction implict (ADI) solving strategy was proposed to predict the temperature field in thin shells. It is particularly adapted to simulate temperature effects in thermo-stamping processes. The main contributions of this work are the following:

• An in-plane/out-of-plane decomposition strategy was proposed. The initial 3D heat transfer problem can be solved in two successive steps:

-solving of a series of 1D problems (

-solving of one 2D problem (

The strong potential of this numerical strategy for computational costs reduc- tion was clearly highlighted.

• The applicability of this solving strategy was investigated. Two conditions are to be fulfilled for the model to be predictive:

-a small aspect ratio for conduction dimensionless ratio

-a small thickness to radius of curvature ratio

These two conditions are fulfilled in standard thermo-stamping industrial cases.

• The proposed formulation is such that the problems

This study is part of the COMMANDO-STAMP project managed by IRT Jules Verne (French Institute in Research and Technology in Advanced Manufactur- ing Technologies for Composite, Metallic and Hybrid Structures). The authors wish to associate the industrial and academic partners of this project; Respec- tively SAFRAN, Peugeot Citroën Automotive, SOLVAY, CEMCAT, LTN, GeM, LAMCOS and 3SR. Also, fruitful discussions with Philippe Boisse and Nahiene Hamila about the integration in a global procedure are to be acknowledged.

Levy, A., Hoang, D.A. and Le Corre, S. (2017) On the Alternate Direction Implicit (ADI) Method for Solving Heat Transfer in Composite Stamp- ing. Materials Sciences and Applications, 8, 37-63. http://dx.doi.org/10.4236/msa.2017.81004

In this Appendix, the surface operator

Mapping. The reference global cartesian system is denoted as

This mapping

Basis. The natural basis at point

and

where the standard comma notation denotes derivation.

Metric tensor. The first fundamental metric tensor of this 2D surface writes, in the local basis,

The unit normal to the tangent surface at point

The second order tensor

and Gaussian curvature

of the surface

Mapping. A position

is defined, where

Because

Basis. At point

Metric tensor. The symmetric fundamental metric tensor

Because the system is parallel curvilinear,

because

Following Equation (64) in [^{1}

^{1}The expression (30) for

where the second order tensor

In the case where the radii of curvature of the surface

and is thus independent of the through thickness position

Gradient. Following [

which can be decomposed, using Equation (32) into an in-plane and an out-of- plane term:

where the surface gradient

In the case where the out-of plane coordinate

As described in section 2.1.2, for a conductivity tensor which has a principal direction in the out of plane direction (Equation (3)), the flux in-plane/out-of- plane decomposition (4) is recovered.

Divergence. First, the following scalar magnitude is defined:

The determinant of

Following [

where the Einstein summation notation is used on the index

where the surface divergence

In the case where the out-of plane coordinate

As given in Section 2.1.2, the heat equation decomposition (5) is recovered.