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Bearing design plays a key role in producing high quality extruded products. Balancing the flow by adjusting the bearing lengths to eliminate profile distortion is a common practice in the industry. The bearing correction, however, is accomplished by costly trial and error approaches – largely relying on the expertise of die designers. In this paper we present a numerical algorithm to optimize the bearing lengths that produce uniform velocity at the die exit. This is based on a finite element model to solve material flow during extrusion. The solution approach involves iteratively computing velocity, temperature, and strain fields during extrusion and updating the bearing lengths until balanced flow is achieved. Different aspects of bearing design and robustness of this optimization technique are illustrated through a few examples.

Modeling the formation of macroscopic segregation channels during directional solidification processes has important applications in the casting industry. Computations that consider thermosolutal convection involve different length scales ranging from the small solute boundary layer at the dendrite tips to the characteristic size of the casting. In general, numerical models of solidification in the presence of a developing mushy zone are computationally inefficient because of nonlinear transport in an anisotropic porous medium. In the current work, mesh adaptation with triangular finite elements is used in conjunction with an efficient fractional-step solver of the momentum equations to predict the occurrence of channel-type segregation defects or freckles. The triangulations are created dynamically using an unstructured grid generator and a refinement criterion that tracks the position of the channel segregates. The efficiency of mesh adaptation is illustrated with simulations showing channel formation and macrosegregation in directional solidification of a Pb–Sn alloy.

Numerical models of solidification including a mushy zone are notoriously inefficient; most of them are based on formulations that require the coupled solution to the velocity components in the momentum equation greatly restricting the range of applicability of the models. Initial attempts at modeling directional solidification in the presence of a developing mushy zone using a projection formulation encountered difficulties once solidification starts, which were traced to the inability of the method to deal with large local density differences in the vicinity of the fluid-mush interface. A modified formulation of the projection method has been developed that maintains the coupling between the body force and the pressure gradient and is presented in this work. This formulation is shown to be robust and extremely efficient; reducing very significantly the necessary storage and the computational time required for the simulation of problems involving very large meshes when compared with previously published data. This is illustrated in this work through its application to simulations involving a Pb-Sn alloy.

Freckles or channel segregates are the most severe form of the macrosegregation that can occur in unidirectionally solidified superalloy castings used in the manufacturing of gas turbine blades. These defects are formed due to thermosolutal convection during solidification. Mathematical modeling of the solidification process involves the simultaneous solution of the conservation equations of momentum, energy and solute concentration in all regions (liquid, mush and solid). Most numerical simulations of dendritic solidification processes have been performed using finite element or the finite volume techniques. The dependence of these methods on the mesh is not always advantageous for problems in which discontinuities or regions of sharp gradients do not coincide with the original mesh lines. In the present work, the meshless element free Galerkin (EFG) method has been investigated to simulate directional solidification processes in which sharp gradients in the field variables can occur as a result of the formation of channels. Simulations of a multicomponent Ni-Al-Ta-W alloy have been performed in a two dimensional domain. The calculations are started with the alloy in all-liquid state and the growth of the mushy zone is followed in time. A projection method is used to solve the momentum equation which makes the computation more efficient than the previously used penalty method. The accuracy of the EFG results is compared with that of the finite element calculations and the potential advantages of the meshless methods for this type of problems are discussed.Copyright © 2009 by ASME

In the present work, the element-free Galerkin (EFG) method is applied to a continuum solidification model that calculates thermosolutal convection and macrosegregation during dendritic solidification of multicomponent alloys. Simulations for directional solidification of a binary Pb-Sn alloy and a Ni-base quaternary alloy have been performed in a rectangular two-dimensional domain. In both calculations, the alloy melt is cooled from below and the growth of the mushy zone is followed in time. The formation of macrosegregation defects known as 'freckles' has been successfully simulated using the meshless EFG method. A varying degree of sensitivity of results to the number and distribution of meshfree particles was obtained. The potential of the method for a broader range of solidification models is discussed.