SIMP Based Topology Optimization for Injection Molding of SFRPs

Abstract

1. Abstract Today there exists a huge demand for technologies which enable and facilitate the mass production of fiber reinforced composites. Injection molding of Short Fiber Reinforced Plastics (SFRP) is a quite popular method especially in the automotive industry, providing high stiffness levels on the one hand and complex moldable shapes on the other hand. Due to the high cost of mold production and injection molding machines, nowadays lots of research is done to improve models and to develop software for the simulation of this process. This allows to detect problems with the mold design and optimization of the part performance and quality at an early stage of the development. In the case of SFRP injection molding, the mechanical properties of the finished part are mainly influenced by the local fiber orientation, which itself depends on the shape/topology of the part. We investigate an approximate approach for compliance-based topology optimization for such parts, where we replace the costly filling and fiber orientation simulation by the solution of an eikonal equation which determines the principal fiber orientation approximately. Sensitivities for compliance minimization are derived for the classical Solid Isotropic Material with Penalization (SIMP) method in combination with a transversely isotropic material law. The optimization problem is then dis-cretized using the Finite Element Method and solved using a projected gradient method. 2. Injection molding of short fiber reinforced thermoplastics is one of the most widely used production processes for the cost efficient mass production of lightweight and strong components. Much research is therefore conducted to optimize this process, mainly focusing on process parameters like melt temperature, fill rate and packing pressure in order to maximize the part quality by reducing warpage and shrinkage effects [9, 26, 23, 18, 4]. Also the optimization of the gate location is often studied in order to obtain a fast balanced filling of the mold [17, 7]. Fewer studies investigate the topology optimization of the mould to reduce weight and cooling time [25] and issues regarding the demoldability of the part [19]. While engineers certainly use topology optimization methods to optimize the stiffness of injection molded components over the conventional trial-and-error approach, no study is known to the authors which includes the dependence of the process induced fiber orientation in the topology optimization specifically for SFRP injection molding. This is certainly owed to the complexity and time consumption of a single injection molding simulation and the un-availability of sensitivities, usually required for an effective optimization. However considering the anisotropy of the fiber orientations becomes especially important for large fiber volume fractions, where the resulting anisotropy of the material is noticeable. In this study we will replace the time consuming fill and fiber orientation simulation by a surrogate model obtained from the solution of an eikonal equation. This enables a fast approximate fiber orientation calculation as well as adjoint based sensitivity analysis for the problem. Of course this surrogate model serves only as preliminary replacement for a full injection molding simulation, as it does not predict the fiber orientations with full accuracy. However, it enables a first investigation of the possible benefits of such a topology optimization. 4. Constitutive Law for SFRP Materials During the fill stage of an injection molded part the fibers are transported and reoriented with the flow. The statistics of the local fiber orientations are usually described in terms of the moments of a fiber orientation density function Ψ : S 2 → R and their evolution are then described by a Fokker-Planck equation. In order to reduce the complexity of the calculations one considers only the 2nd and 4th moments of the fiber orientation density function [24]