Magnesium (Mg) alloys are the lightest commercially available structural metals, with a density of 1.74 \(g/{cm}^{3}\). In recent years, they have attracted much attention for vehicle light-weighting applications [1]. While Mg alloys hold great promise in automotive applications – owing to their high strength and stiffness-to-weight ratios, castability, machinability, and damping [2] – much of their use has been limited to the production of non-load-bearing components due to inferior cast properties and poor formability resulting from a strong crystallographic texture [3]. Casting has been the most dominant manufacturing process for producing Mg components via methods such as high-pressure die casting (HPDC), gravity, sand, and permeant mould casting (PMC) [4]. While casting processes can offer greater design flexibility, the cast products can exhibit coarse and non-uniform microstructure and porosity defects. These drawbacks limit the use of casting as a primary method for manufacturing structurally demanding, load-bearing components. Plastic deformation of the cast structure can mitigate these issues as the as-cast grain structure becomes refined, improving the strength and ductility of the deformed structure. Hot forging is a manufacturing process where plastic deformation is induced in the workpiece. Hot forged Mg alloy components exhibit superior mechanical properties compared to their cast counterpart as the strain energy that is imposed via forging refines the structure of the workpiece into a more uniform and fine-grained structure. Typically, during hot forging, a wrought or cast billet is subjected to one or more pre-forming steps to forge intermediate shapes prior to the final forge. Naturally, the extra tooling that is required increases the costs associated with the process. This is one of the reasons forged Mg alloy products have been confined to high-cost sports and military applications [5].
Typically, the preform design of a forging operation is conceived via a trial-and-error approach where an engineer would iteratively re-model and analyze the preform deformation behavior using finite element method (FEM) simulation tools. Such an approach limits the space of designs that can be explored and can lead to a sub-optimal solution depending on the set of heuristics that were considered. In the past, to overcome this issue, several systematic shape optimization methods have been proposed to help guide the direction of shape exploration. The backward tracing scheme (BTS) is one such method where the metal flow simulation is reversed starting from a desired final configuration of the forged shape. At each backward step, the bounding nodes of the preform at the die interface are released based on their velocities; the nodes with the smallest velocities have the least influence on deformation. The problem is solved iteratively until all but one of the nodes have been separated from the die and the preform shape is obtained which satisfies a single objective such as minimization of variation of effective strain within the forged component or flash volume [6]–[9]. While BTS has been successfully implemented by many researchers, it is not a trivial task to define a good nodal separation criterion, especially in complex 3D shape optimization problems. A topology optimization method known as bidirectional evolutionary structural optimization (BESO) is more suited for such problems. This method involves iteratively adding and removing elements from a voxelized mesh of the preform at each simulation step based on either a hydrostatic stress-based or strain-based criterion [10]. This modified voxel mesh is then subjected to a surface smoothing operation to obtain a valid mesh that can be used in FEM simulations [11]. To apply BESO in complex 3D shape optimization problems, several task-specific factors such as the sub-routines for 3D element addition and removal, preform-die interference checking, and mesh smoothing, need to be established. In addition, like BTS optimization, it is difficult to apply BESO to obtain solutions that confer multi-objective advantages. For multi-objective shape optimization problems, researchers have used surrogate models in conjunction with evolutionary algorithms (EAs) to evolve preform shapes. A surrogate model in this context would typically be developed using a subset of designs that cover the design space with the intended purpose of approximating preform shape performance (i.e., the forging outcome) which would otherwise be evaluated using computationally expensive FEM simulations. Then, an EA such as a genetic algorithm (GA) would be used to explore the space of preform designs for an optimal solution [12], [13], or a set of Pareto optimal solutions if multiple objectives were to be achieved [14], [15]. EAs confer several advantages over gradient-based optimization methods such as the ability to find global optima, robustness to noisy input data, the ability to handle non-smooth or discontinuous fitness functions, and the ability to incorporate multiple fitness functions simultaneously [16]. Using this approach, Torabi et al. successfully developed a response surface model with a multi-objective GA to optimize the preform shape of a 3D blade to achieve a maximum filling ratio, minimum flash volume, forging load, and strain variance [17]. Similarly, Shao et al. [18] used a radial basis function model with a multi-island genetic algorithm (MIGA) to optimize the preform shape of an airfoil to improve forging quality. While surrogate models and EAs have been used extensively in the past, there is a noticeable gap in research that apply these methods in multi-objective shape optimization problems of more geometrically complex 3D shapes. More specifically, there is a lack of research on effective methodologies for rapidly generating 3D models. Additionally, the surrogate models that have been developed only give a global evaluation of the forging outcome – they do not capture forging outcome in spatially varying regions of interest.
This work is part of a larger research effort to establish the required scientific knowledge and technology to combine casting and forging as a sequence of manufacturing steps (cast-forging) for producing Mg-alloy components that are both cost-effective and optimized for structural applications. The focus of this paper is to introduce a multi-objective optimization framework and its application towards obtaining an optimal Mg-alloy preform design to produce an I-beam forging. This simplified I-beam geometry was used throughout the program to act as a surrogate for a more complex automotive lower control arm. The proposed framework consists of a parametric computer-aided design (CAD) model for shape generation [19], data-driven models for shape evaluation, and a multi-objective evolutionary algorithm for design space search. The framework is applied to obtain a preform design that outperforms the forging outcome of a baseline, cast-cylindrical billet.