The high-speed machining process is a dynamic one in which material is removed from bulk stock through the motion of a tool moving at very high speeds over the surface. The process converts mechanical energy into thermal energy due to both the bulk pastic response of the workpiece and the frictional heating at the tool-chip surface. This project is a collaboration between Duke University, the University of Notre Dame, and Alcoa. Researchers at Notre Dame (directed by Professor Jim Mason) are conducting experimental studies of the temperature distribution in the workpiece. Duke is responsible for the simulation efforts.

We are investigating both transient and steady-state cutting conditions with numerical methods. The steady-state formulation employs the X-FEM to update the chip geometry without remeshing, and employs mortar elements to impose contact constraints. The transient problem is being studied with the commercial software D-FORM. This project is sponsored by the National Science Foundation.

Contour plot of the temperature distribution (in degrees Celsius) in an aluminum workpiece being cut at a speed of 50m/s.

The Extended Finite Element Method is a new technique for the finite element modelling of internal features such as material interfaces, sliding interfaces, and cracks. This method allows for the modelling of arbitrary moving geometric features independently of the finite element mesh with evolving enrichment functions. Especially, this method has an advantage to model crack growth without remeshing. The main idea is to extend a classical approximation by augmenting the set of nodal shape functions with products of a subset of these same functions and local enrichment functions. The following figures show the difference of the standard finite element mesh and the extended finite element mesh and the results for a crack growth.

Contour plot of the stress distribution of FEM mesh(above) and XFEM mesh(below)

The level set method is to track moving interfaces in a wide variety of prolbems. It relies of the relation between propagation interfaces and propagating shocks. The central idea is to follow the evolution of a function whose zero level set always corresponds to the position of the propagating interface. The motion for this evolving function is determined from a partial differential equation in one higher dimension which permits cusps, sharp corners, and changes in topology in the zero level set describing the interface. The level set approach has been used to compute and analyze a broad array of physical and mathematical phenomena, including singularities in mean curvature flow, crystal growth and dendrite solidification, combustion, shape recognition, minimal surface generation, two fluid problems, phase transition, modelling of material interfaces, and triple junction problems. I am using this technique to track a moving chip geometry during the machining process. The following figure is the signed distance function used to initialize the level set partial differential equation.

The mortar element method is to deal with non-conforming discretizations across contacting boundaries. In those contact problems where conforming discretizations are to be expected, parametrization of the contact surfaces is straightforward. For example, when one deformable body is subject to unilateral contact constraints imposed by a rigid obstacle, the contact surface can inherit element discretizations along the contacting element edges without any ambiguity or subsequent difficulty in numerical approximation. Likewise, in small deformation contact problems where the meshes of the contacting surfaces are conforming, the contact quantities can be taken to vary according to the shared element parametrizations along the surface. However, in large deformation two-body problems, the distortion of elements comprising the surface and their potenrially large relative motion make it impractical to maintain conforming discretization across the contact surface.
Many efforts have circumvented the surface discretization issue in large deformations by considering variants of the so-called master/slave strategy, whereby the nodes of one deformable surface (the slave surface) are prohibited from penetrating the element edges of the opposing surface (the master surface). This approach would appear to be pervasive in commercial finite element packages currently. Such an approach reflects a primary interest in the physical requirement of impenetrability, enforcing the constraint at nodal points and calculating the nodal forces necessary to maintain impenetrability at these points while maintaining global equilibrium. However, such approaches are often incapable of passing so-called contact patch tests, wherein a flat contact surface is called upon to transmit a spatially constant pressure from one body to the other.