**1. Introduction**

Composite reinforced by woven or UD fabric is known to have high specific stiffness and, in combination with automatic manufacturing processes, makes it possible to fabricate complex components in various industry sectors (aircraft, boat, automotive, and military). Over the last years, the demand for high stiffness and strength and low-weight materials, such as fiberreinforced plastics, has grown in the transport industry. Especially in the aeronautical industry, the use of woven-fabric-reinforced plastics has increased significantly. The main objective of aerospace industries is to reduce to half the amount of fuel by 2020 and at least 70% less by 2025. Composite manufacturing processes have undergone substantial evolution in recent years [1, 2]. Although the traditional layup process will remain the process of choice for some applications, new developments in Resin Transfer Molding (RTM), Liquid Composite Molding (LCM), or Sheet Molding Compound (SMC), low-temperature curing pre-pregs and low-pressure molding compounds have matured significantly, and are now being exploited in high-technology areas such as aerospace and automotive industries [3-7].

The manufacturing of reinforced composites needs different forming step in which the preform fabric (woven or nonwoven) takes the desired product's shape. The main deformation

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mechanism during forming of woven or nonwoven-reinforced composites is shear, which causes a change in fiber orientations. Fiber reorientation is one of the major factors causing fabric distortions, shrinking, and warpage defect. The fiber reorientation is an important factor that should be taken into account when designing composite products, since it will influence the overall thermomechanical properties and performance.

In this context, numerical simulation methods are needed to anticipate the performance of the final product, but also to predict the reinforcement preforming and the resin injection. Several modeling approaches have been developed in the literature to account for the evolution of the fiber orientation [8-11]. The earliest technique is based on discrete mapping approaches. In contrast to these mapping schemes, the constitutive behavior is required for continuum mechanical approaches [9, 12, 13].

The mapping approach, the so-called kinematic method, is used to determine the de‐ formed shape of draped fabrics. The main assumptions are that the warp and weft fibers are inextensible, intersection points between warp and weft yarns are fixed during preforming, and the angle between warp and weft yarns are free. This method, where the fabric is placed progressively from an initial line, provides a close enough resemblance to hand-made draping [14-17].

The alternative to the kinematic approach consists of the use of Finite Element (FE) methods to simulate the fabric deformation under the boundary conditions prescribed by the forming process by considering the fabric as a homogeneous material using computationally efficient constitutive laws and continuum FEs. The limitation of the FE method is that the fabric is not really a continuum but can be more closely likened to a structure comprising discrete rods, possibly intertwined (for woven fabric), or loosely held together with stitching (for nonwoven fabric). The draping of composite fabric using a mechanical approach requires the resolution of equilibrium PDE's problems by the FE method. In general, in the case of complex surfaces, the boundary conditions are not well-defined and the contact between the surface and the fabric is difficult to manage [18, 19, 20]. Furthermore, the resolution of such a problem can be too long in CPU time and is detrimental to the optimization stage of draping regarding the initial fiber directions. All of these facts lead us to consider rather a kinematic approach, which is very fast and more robust, allowing simultaneously to define the stratification sequences and the flat pattern for different plies and to predict difficult impregnated areas that involve manual operation like dart insertion or, on the contrary, the shortage of fabric [21-38].

This paper presents an optimization-based method for simulation of forming processes of woven and nonwoven fabric reinforced composites using geometrical approach. Two draping simulation examples are given. These simulations are performed using the geometrical analysis computer code. For each example, we assume that a mesh of the mold to drape is given. The first example is the draping of woven fabric on double dome mold geometry. The second example shows the influence of the woven fabric and nonwoven on the draping process. The effects of the initial conditions (fiber orientation and start point) on the draped preform are discussed.
