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 Finite Element Methods for Twin-Screw Extruders: Unsteady - Temperature-Dependent - Non-Newtonian Simulations

We present a boundary-conforming space-time finite element method to compute the flow inside co-rotating, self-wiping twin-screw extruders. The mesh update is carried out using the newly developed Snapping Reference Mesh Update Method (SRMUM). It allows to compute time-dependent flow solutions inside twin-screw extruders equipped with conveying screw elements without any need for re-meshing and projections of solutions - making it a very efficient method. We provide cases for Newtonian and non-Newtonian fluids in 2D and 3D, that show mesh convergence of the solution as well as agreement to experimental results. Furthermore, a complex, unsteady and temperature-dependent 3D test case with multiple screw elements illustrates the potential of the method also for industrial applications

Fully-implicit log-conformation formulation of constitutive laws

Subject of this paper is the derivation of a new constitutive law in terms of the logarithm of the conformation tensor that can be used as a full substitute for the 2D governing equations of the Oldroyd-B, Giesekus and other models. One of the key features of these new equations is that - in contrast to the original log-conf equations given by Fattal and Kupferman (2004) - these constitutive equations combined with the Navier-Stokes equations constitute a self-contained, non-iterative system of partial differential equations. In addition to its potential as a fruitful source for understanding the mathematical subtleties of the models from a new perspective, this analytical description also allows us to fully utilize the Newton-Raphson algorithm in numerical simulations, which by design should lead to reduced computational effort. By means of the confined cylinder benchmark we will show that a finite element discretization of these new equations delivers results of comparable accuracy to known methods.Comment: 21 pages, 5 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

Combining Boundary-Conforming Finite Element Meshes on Moving Domains Using a Sliding Mesh Approach

For most finite element simulations, boundary-conforming meshes have significant advantages in terms of accuracy or efficiency. This is particularly true for complex domains. However, with increased complexity of the domain, generating a boundary-conforming mesh becomes more difficult and time consuming. One might therefore decide to resort to an approach where individual boundary-conforming meshes are pieced together in a modular fashion to form a larger domain. This paper presents a stabilized finite element formulation for fluid and temperature equations on sliding meshes. It couples the solution fields of multiple subdomains whose boundaries slide along each other on common interfaces. Thus, the method allows to use highly tuned boundary-conforming meshes for each subdomain that are only coupled at the overlapping boundary interfaces. In contrast to standard overlapping or fictitious domain methods the coupling is broken down to few interfaces with reduced geometric dimension. The formulation consists of the following key ingredients: the coupling of the solution fields on the overlapping surfaces is imposed weakly using a stabilized version of Nitsche's method. It ensures mass and energy conservation at the common interfaces. Additionally, we allow to impose weak Dirichlet boundary conditions at the non-overlapping parts of the interfaces. We present a detailed numerical study for the resulting stabilized formulation. It shows optimal convergence behavior for both Newtonian and generalized Newtonian material models. Simulations of flow of plastic melt inside single-screw as well as twin-screw extruders demonstrate the applicability of the method to complex and relevant industrial applications

Boundary-Conforming Finite Element Methods for Twin-Screw Extruders using Spline-Based Parameterization Techniques

This paper presents a novel spline-based meshing technique that allows for usage of boundary-conforming meshes for unsteady flow and temperature simulations in co-rotating twin-screw extruders. Spline-based descriptions of arbitrary screw geometries are generated using Elliptic Grid Generation. They are evaluated in a number of discrete points to yield a coarse classical mesh. The use of a special control mapping allows to fine-tune properties of the coarse mesh like orthogonality at the boundaries. The coarse mesh is used as a 'scaffolding' to generate a boundary-conforming mesh out of a fine background mesh at run-time. Storing only a coarse mesh makes the method cheap in terms of memory storage. Additionally, the adaptation at run-time is extremely cheap compared to computing the flow solution. Furthermore, this method circumvents the need for expensive re-meshing and projections of solutions making it efficient and accurate. It is incorporated into a space-time finite element framework. We present time-dependent test cases of non-Newtonian fluids in 2D and 3D for complex screw designs. They demonstrate the potential of the method also for arbitrarily complex industrial applications

Computing the jump-term in space-time FEM for arbitrary temporal interpolation

[EN] One approach with rising popularity in analyzing time-dependent problems in scienceand engineering is the so-called space-time finite-element method that utilized finiteelementsin both space and time. A common ansatz in this context is to divide the meshin temporal direction into so-called space-time slabs, which are subsequently weaklyconnected in time with a Discontinuous Galerkin approach. The corresponding jumpterm,which is responsible for imposing the weak continuity across space-time slabs canbe challenging to compute, in particular in the context of deforming domains. Ensuringa conforming discretization of the space-time slab at the top and bottom in timedirection simplifies the handling of this term immensely. Otherwise, a computationallyexpensive and error prone projection of the solution from one time-level to another isnecessary. However, when it comes to simulations with deformable domains, e.g. forfree-surface flows, ensuring conforming meshes is quite laborious. A possible solutionto this challenge is to extrude a spatial mesh in time at each time-step resulting in theso-called time-discontinuous prismatic space-time (D-PST) method[1]. However, thisprocedure is restricted to finite-elements of 1st order in time.We present a novel algorithmic approach for arbitrarily discretized meshes by flippingthe mesh in time-direction for each time-step. This ansatz allows for a simple evaluationof the jump-term as the mesh is always conforming. The cost of flipping the mesharound its symmetry plane in time scales with the number of nodes, which makes itcomputationally cheaper than an additional update of the mesh to enforce conformity orthe evaluation of a projection. We validate the approach on various physical problemswith and without deforming domains.Salzmann, E.; Zwickey, F.; Elgeti, S. (2022). Computing the jump-term in space-time FEM for arbitrary temporal interpolation. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 223-231. https://doi.org/10.4995/YIC2021.2021.12588OCS22323

A Space-Time FE Level-set method for convection coupled phase-change processes

[EN] Phase transition processes have great relevance for both engineering and scientific applications. In production engineering, for instance, metal welding and alloy solidification are topics of ongoing research.In this contribution we focus on the convection coupled solid-liquid phase change of a single species, e.g. water. The material is assumed to be incompressible within the two phases, but we account for density changes across the phase interface. To describe the process, we need to solve the incompressible Navier-Stokes equations and the heat equation for both phases over time. The position of the phase interface is tracked with a Level-set method. The Level-set function is advected according to the propagation speed of the phase interface. Such velocity field depends on local energy conservation across the interface and is modelled as the Stefan condition. This formulation requires us to approximate the heat flux discontinuity across the interface based on the evolving temperature and velocity fields.To model the temperature and velocity fields within each phase, we employ the Space-Time Finite Element method. However, commonly used interpolation functions, such as piecewise linear functions, fail to capture discontinuous derivatives over one element that are needed to assess the Level-set's transport term. Available solutions to this matter, such as local enrichment with Extended Finite Elements, are often not compatible with existing Space-Time Finite Element codes and require extensive implementation work. Instead, we consider a conceptually simpler method and we decide to extend the Ghost Cell technique to Finite Element meshes. The idea is that we can separate the two subdomains associated with each phase and solve two independent temperature problems. We prescribe the melting temperature at an additional node close to the interface and we retrieve the required heat flux.In this work we describe the Ghost Cell method applied to our Space-Time Finite Element solver. First, we verify numerical results against analytical solutions, then we demonstrate more complex test cases in 2D and 3D.The authors were supported by the Helmholtz Graduate School for Data Science in Life, Earth and Energy (HDS-LEE). The work was furthermore supported by the Federal Ministry of Economic Affairs and Energy, on the basis of a decision by the German Bundestag (50 NA 1908). The authors gratefully acknowledge the computing time granted by the JARA Vergabegremium and provided on the JARA Partition part of the supercomputer JURECA at Forschungszentrum Jülich .Boledi, L.; Terschanski, B.; Elgeti, S.; Kowalski, J. (2022). A Space-Time FE Level-set method for convection coupled phase-change processes. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 206-213. https://doi.org/10.4995/YIC2021.2021.12329OCS20621