Analysis of Solidification and Viscoplastic Stresses Incorporating a Moving Boundary: An Application to Simulation of the Centrifugal Casting Process

An efficient finite element scheme for viscoelasticity with moving boundaries

“…Now we consider the global semi-discrete system (13). First we observe that the vector Zt n cannot be written in the form (14) in the moving-mesh case. The reason is the following.…”

“…The reason is the following. In (14), the history matrix Ht À s multiplies the vector Us, which is supposed to represent the solution history at ®xed locations. However, in the movingmesh case, the location of the nodal points changes in time, and thus Us tracks the history at the moving nodal points rather than representing the history at ®xed points!…”

“…The simplest integration rule for (14) is the trapezoidal rule, with intervals of size Dt [25]. The integral term (14) in the system (13) has a hereditary nature; it involves a convolution in time and appears due to the``memory'' property associated with the viscoelastic constitutive behavior (see (2)). Equation (14) implies that at any time t n , the solution process involves past values of Ut, from 0 to the current time t n .…”

“…However, for a particular (but important) class of relaxation functions, namely those which can be written as (or approximated by) a series of exponentials, a wellknown``memory trick'' can be applied which circumvents this dif®culty, and which allows a very ef®cient calculation of Zt n in (14). The trick has been ®rst proposed by Herrmann and Peterson [21] and Taylor et al [22], and has been frequently used.…”

“…See, e.g., [13,14] for fully coupled analyses. In uncoupled thermomechanical problems, the thermal ®eld and the motion of the boundary are found ®rst, and as a second step, based on these results, the deformation and stresses are solved for.…”