Inverse Design Based on Nonlinear Thermoelastic Material Models Applied to Injection Molding

This paper describes an inverse shape design method for thermoelastic bodies. With a known equilibrium shape as input, the focus of this paper is the determination of the corresponding initial shape of a body undergoing thermal expansion or contraction, as well as nonlinear elastic deformations. A distinguishing feature of the described method lies in its capability to approximately prescribe an initial heterogeneous temperature distribution as well as an initial stress field even though the initial shape is unknown. At the core of the method, there is a system of nonlinear partial differential equations. They are discretized and solved with the finite element method or isogeometric analysis. In order to better integrate the method with application-oriented simulations, an iterative procedure is described that allows fine-tuning of the results. The method was motivated by an inverse cavity design problem in injection molding applications. Its use in this field is specifically highlighted, but the general description is kept independent of the application to simplify its adaptation to a wider range of use cases.Comment: 22 pages, 13 figure

Boundary-Conforming Free-Surface Flow Computations: Interface Tracking for Linear, Higher-Order and Isogeometric Finite Elements

The simulation of certain flow problems requires a means for modeling a free fluid surface; examples being viscoelastic die swell or fluid sloshing in tanks. In a finite-element context, this type of problem can, among many other options, be dealt with using an interface-tracking approach with the Deforming-Spatial-Domain/Stabilized-Space-Time (DSD/SST) formulation. A difficult issue that is connected with this type of approach is the determination of a suitable coupling mechanism between the fluid velocity at the boundary and the displacement of the boundary mesh nodes. In order to avoid large mesh distortions, one goal is to keep the nodal movements as small as possible; but of course still compliant with the no-penetration boundary condition. Standard displacement techniques are full velocity, velocity in a specific coordinate direction, and velocity in normal direction. In this work, we investigate how the interface-tracking approach can be combined with isogeometric analysis for the spatial discretization. If NURBS basis functions of sufficient order are used for both the geometry and the solution, both a continuous normal vector as well as the velocity are available on the entire boundary. This circumstance allows the weak imposition of the no-penetration boundary condition. We compare this option with an alternative that relies on strong imposition at discrete points. Furthermore, we examine several coupling methods between the fluid equations, boundary conditions, and equations for the adjustment of interior control point positions.Comment: 20 pages, 16 figure

Automatic implementation of material laws: Jacobian calculation in a finite element code with TAPENADE

In an effort to increase the versatility of finite element codes, we explore the possibility of automatically creating the Jacobian matrix necessary for the gradient-based solution of nonlinear systems of equations. Particularly, we aim to assess the feasibility of employing the automatic differentiation tool TAPENADE for this purpose on a large Fortran codebase that is the result of many years of continuous development. As a starting point we will describe the special structure of finite element codes and the implications that this code design carries for an efficient calculation of the Jacobian matrix. We will also propose a first approach towards improving the efficiency of such a method. Finally, we will present a functioning method for the automatic implementation of the Jacobian calculation in a finite element software, but will also point out important shortcomings that will have to be addressed in the future.Comment: 17 pages, 9 figure

Inverse shape design in injection molding based on the finite element method

A method is proposed for the inverse design of cavity shapes for the injection molding process. When liquid polymer melt is cooled down in an injection mold to manufacture plastics parts, inhomogeneities in the cooling and solidification processes lead to shape defects in the finished molding. The geometry of the cavity where the liquid melt is injected is largely responsible for the shape of the molding. The method described in this document offers an automatized tool for the determination of a suitable cavity shape that will reduce faults in the molding shape. The basis of this method is a numerical simulation of the injection molding process. This method builds on simulation models for both fluid and solid polymers that incorporate the important physical phenomena of thermoviscoelastic material behavior and solidification. Separate simulation models are described in this document for the solidification and post-ejection stages of the process. They are both equipped with a finite element formulation that makes them suitable for a swift implementation in a computer code. The inverse design method for the cavity shape results from a combination of an inverse formulation of stationary thermoelasticity with an iteration scheme that incorporates the non-elastic effects. This iterative method is demonstrated for two sample cases. The simulation method is shown to represent the important aspects of the viscoelastic behavior and solidification. The iterative inverse design method produces suitable cavity shapes after small numbers of iteration steps. Furthermore, plots of a distance measure over the course of the iteration indicate rapid convergence of the method