A condensed presentation of some results of a geometric character, concerning the injection molding of plastics, is given here. The author has derived these results from mathematical arguments and some simplifying assumptions, besides the usual Hele-Shaw flow conditions.
The presentation here is intended for readers with an interest in polymer processing, rather than mathematics, so that the mathematical derivations are omitted in some cases, and sketchy in other cases. Instead we try to explain the results using figures, intuitive arguments and a few inevitable formulas. Since the experimental verification of the results is still very incomplete, we prefer to present them as proposed principles. Comments and suggestions for improvement are very welcome.
This paper deals with the interpolation of given real boundary values into a bounded domain in Euclidean n-space, under a prescribed gradient bound. It is well known that there exist an upper solution (ail inf-convolution) and a lower solution (a sup-convolution) to this problem, provided that a certain compatibility condition is satisfied. If the upper and lower solutions coincide somewhere in the domain, then several interesting consequences follow. They are considered here. Basically, the upper and lower solutions must be regular wherever they coincide.
This paper deals with minimax problems for nonlinear differential expressions involving a vector-valued function of a scalar variable under rather conventional structure conditions on the cost function. It is proved that an absolutely minimizing (i.e. globally and locally minimizing) function is continuously differentiable. A minimizing function is also continuously differentiable, provided a certain extra condition is satisfied. The variational method of V.G. Boltyanskii, developed within optimal control theory, is adapted and used in the proof. The case of higher order derivatives is also considered.
Consider a company which produces and sells a certain product on a market with highly variable demand. Since the demand is very high during some periods, the company will produce and create a stock in advance before these periods. On the other hand it costs money to hold a big stock, so that some balance is needed for optimum. The demand is assumed to be known in advance with sufficient accuracy. We use a technique from optimal control theory for the analysis, which leads to so-called activity periods. During such a period the stock is positive and the production is maximal, provided that the problem starts with zero stock, which is the usual case. Over a period of one or more years, there will be a few activity periods. Outside these periods the stock is zero and the policy is to choose production = the smaller of [demand, maximal production]. The “intrinsic time length” is a central concept. It is simply the maximal time a unit of the product can be stored before selling without creating a loss.
Various approaches are used to derive the Aronsson-Euler equations for L-infinity calculus of variations problems with constraints. The problems considered involve holonomic, nonholonomic, isoperimetric, and isosupremic constraints on the minimizer. In addition, we derive the Aronsson-Euler equation for the basic L-infinity problem with a running cost and then consider properties of an absolute minimizer. Many open problems are introduced for further study.
These notes are intended to be a rather complete and self-contained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowledge of the subject. In particular, we improve known results regarding existence via arguments that are simpler than those that can be found in the literature. We present a proof of the main known uniqueness result which is largely self-contained and does not rely on the theory of viscosity solutions. A unifying idea in our approach is the use of cone functions. This elementary geometric device renders the theory versatile and transparent. A number of tools and issues routinely encountered in the theory of elliptic partial differential equations are illustrated here in an especially clean manner, free from burdensome technicalities - indeed, usually free from partial differential equations themselves. These include a priori continuity estimates, the Harnack inequality, Perron's method for proving existence results, uniqueness and regularity questions, and some basic tools of viscosity solution theory. We believe that our presentation provides a unified summary of the existing theory as well as new results of interest to experts and researchers and, at the same time, a source which can be used for introducing students to some significant analytical tools.
A detailed analysis of the class of absolutely minimizing functions in Euclidean spaces and the relationship to the infinity Laplace equation
We discuss an idealized model for compression molding, had by taking an asymptotic limit for highly non-Newtonian materials. We interpret the changing pressure distributions as being dictated by a Monge-Kantorovich mass transfer on a fast time scale, and thereby derive a nonlocal geometric law of motion for the air/plastic interface.
The simulation of the advancing flow front during mould filling in the injection moulding cycle is important as a way of anticipating manufacturing defects, particularly different types of welds, air traps, cold spots and hot spots. The purpose of the present work was to evaluate a model (the distance model) simulating the advancing flow front and predicting potential issues related to the progression of the flow front such as welds and air traps. The distance model is based on a mathematical theory of Hele-Shaw flow for strongly shear-thinning fluids, i.e. fluids with a power-law index, n, equal to 0,3 or preferably less. Two grades of general-purpose polystyrene were selected according to their abilities to shear thin (weakly or moderately strong) at the selected processing temperature and in the typical shear rate range of the injection moulding process. The two grades were injection moulded into twelve mould configurations derived from two similar single-gated moulds. The flow length from the gate to the weld was measured in the mouldings obtained at three distinct rates of filling (conventional, slower and faster) and compared to the flow lengths obtained by the distance model. Good agreement was found between the predicted flow lengths and the experimental flow lengths at the conventional and faster filling conditions. Larger deviations between simulation and experiment were found at the slower filling rate, particularly for the weakly shear-thinning polystyrene grade. Some comparisons were also made with the predictions obtained using the commercial simulation code Moldflow. A comparison between the advancing front predictions of the distance model and experimental short shots of a commercial polypropylene grade in a lure-box type of mould geometry showed that the distance model, despite its simplicity, could probably be used to detect welds and air traps in more complex and practice-related mouldings.