Stefanie Elgeti

Norbert Hosters, Jan Helmig, Atanas Stavrev et al.

Engineering design via CAD software relies on Non-Uniform Rational B-Splines (NURBS) as a means for representing and communicating geometry. Therefore, in general, a NURBS description of a given design can be considered the exact description. The development of isogeometric methods has made the geometry available to analysis methods Hughes et al. (2005). Isogeometric analysis has been particularly successful in structural analysis; one reason being the wide-spread use of two-dimensional finite elements in this field. For fluid dynamics, where three-dimensional analysis is usually indispensable, isogeometric methods are more complicated, yet of course not impossible, to apply in a general fashion. This paper describes a method that enables the solution of fluid-structure interaction with a matching spline description of the interface. On the structural side, the spline is used in an isogeometric setting. On the fluid side, the same spline is used in the framework of a NURBS-enhanced finite element method (extension of Sevilla et al. (2011)). The coupling of the structural and the fluid solution is greatly facilitated by the common spline interface. The use of the identical spline representation for both sides permits a direct transfer of the necessary quantities, all the while still allowing an adjusted, individual refinement level for both sides.

Stefanie Elgeti, Henning Sauerland

Fluid flow applications can involve a number of coupled problems. One is the simulation of free-surface flows, which require the solution of a free-boundary problem. Within this problem, the governing equations of fluid flow are coupled with a domain deformation approach. This work reviews five of those approaches: interface tracking using a boundary-conforming mesh and, in the interface capturing context, the level-set method, the volume-of-fluid method, particle methods, as well as the phase-field method. The history of each method is presented in combination with the most recent developments in the field. Particularly, the topics of extended finite elements (XFEM) and NURBS-based methods, such as Isogeometric Analysis (IGA), are addressed. For illustration purposes, two applications have been chosen: two-phase flow involving drops or bubbles and sloshing tanks. The challenges of these applications, such as the geometrically correct representation of the free surface or the incorporation of surface tension forces, are discussed.

Violeta Karyofylli, Markus Frings, Stefanie Elgeti et al.

In this paper, we present the numerical solution of two-phase flow problems of engineering significance with a space-time finite element method that allows for local temporal refinement. Our basis is the method presented in [3], which allows for arbitrary temporal refinement in preselected regions of the mesh. It has been extended to adaptive temporal refinement that is governed by a quantity that is part of the solution process, namely, the interface position in two-phase flow. Due to local effects such as surface tension, jumps in material properties, etc., the interface can, in general, be considered a region that requires high flexibility and high resolution, both in space and in time. The new method, which leads to tetrahedral (for 2D problems) and pentatope (for 3D problems) meshes, offers an efficient yet accurate approach to the underlying two-phase flow problems.

Florian Zwicke, Philipp Knechtges, Marek Behr et al.

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.

Fabian Key, Lutz Pauli, Stefanie Elgeti

A novel method - the Virtual Ring Shear-Slip Mesh Update Method (VR-SSMUM) - for the efficient and accurate modeling of moving boundary or interface problems in the context of the numerical analysis of fluid flow is presented. We focus on cases with periodic straight-line translation including object entry and exit. The periodic character of the motion is reflected in the method via a mapping of the physical domain onto a closed virtual ring. Therefore, we use an extended mesh, where unneeded portions are deactivated to control the computational overhead. We provide a validation case as well as examples for the applicability of the method to 2D and 3D models of packaging machines.

Clemens Fricke, Daniel Wolff, Marco Kemmerling et al.

Profile extrusion is a continuous production process for manufacturing plastic profiles from molten polymer. Especially interesting is the design of the die, through which the melt is pressed to attain the desired shape. However, due to an inhomogeneous velocity distribution at the die exit or residual stresses inside the extrudate, the final shape of the manufactured part often deviates from the desired one. To avoid these deviations, the shape of the die can be computationally optimized, which has already been investigated in the literature using classical optimization approaches. A new approach in the field of shape optimization is the utilization of Reinforcement Learning (RL) as a learning-based optimization algorithm. RL is based on trial-and-error interactions of an agent with an environment. For each action, the agent is rewarded and informed about the subsequent state of the environment. While not necessarily superior to classical, e.g., gradient-based or evolutionary, optimization algorithms for one single problem, RL techniques are expected to perform especially well when similar optimization tasks are repeated since the agent learns a more general strategy for generating optimal shapes instead of concentrating on just one single problem. In this work, we investigate this approach by applying it to two 2D test cases. The flow-channel geometry can be modified by the RL agent using so-called Free-Form Deformation, a method where the computational mesh is embedded into a transformation spline, which is then manipulated based on the control-point positions. In particular, we investigate the impact of utilizing different agents on the training progress and the potential of wall time saving by utilizing multiple environments during training.

Steffen Tillmann, Felipe A. González, Stefanie Elgeti

Purpose: In fused deposition modeling (FDM), the nozzle plays a critical role in enabling high printing speeds while maintaining precision. Despite its importance, most applications still rely on standard nozzle designs. This work investigates the influence of nozzle geometry on pressure loss inside the nozzle, a key factor in high-speed printing performance. Design/methodology/approach: We focus on optimizing the nozzle shape to minimize the pressure loss and establish a framework that allows both simple angle-based optimization and more advanced spline-based parametrization. To model the polymer melt flow, we use a Giesekus model to account for viscoelastic effects. Findings: For angle-based optimization, the pressure-loss objective exhibits two local minima: one associated with smooth flow and another with pronounced recirculation regions inside the nozzle. While the latter yields a lower pressure drop, such flow patterns are generally undesirable due to increased residence times and the associated risk of material degradation and nozzle clogging. The splinebased parametrization results in only marginal additional reductions in pressure loss compared to angle optimization, while decreasing the manufacturability of the nozzle considerably. Originality/value: This paper presents a comparative study of FDM nozzle shape optimization using a Giesekus model. We introduce a flexible optimization framework that accommodates both simple and advanced geometric parametrizations. The main contribution is the systematic comparison between angle- and spline-based parametrizations across materials and extrusion velocities, showing that most of the achievable pressure-loss reduction is already captured by the simpler and more manufacture-ready angle optimization.

Michel Make, Norbert Hosters, Marek Behr et al.

This article considers the NURBS-Enhanced Finite Element Method (NEFEM) applied to the compressible Navier-Stokes equations. NEFEM, in contrast to conventional finite element formulations, utilizes a NURBS-based computational domain representation. Such representations are typically available from Computer-Aided-Design tools. Within the NEFEM, the NURBS boundary definition is utilized only for elements that are touching the domain boundaries. The remaining interior of the domain is discretized using standard finite elements. Contrary to isogeometric analysis, no volume splines are necessary. The key technical features of NEFEM will be discussed in detail, followed by a set of numerical examples that are used to compare NEFEM against conventional finite element methods with the focus on compressible flow.

Florian Zwicke, Sebastian Eusterholz, Stefanie Elgeti

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.

Brendan Keith, Philipp Knechtges, Nathan V. Roberts et al.

We explore a vexing benchmark problem for viscoelastic fluid flows with the discontinuous Petrov-Galerkin (DPG) finite element method of Demkowicz and Gopalakrishnan [1,2]. In our analysis, we develop an intrinsic a posteriori error indicator which we use for adaptive mesh generation. The DPG method is useful for the problem we consider because the method is inherently stable---requiring no stabilization of the linearized discretization in order to handle the advective terms in the model. Because stabilization is a pressing issue in these models, this happens to become a very useful property of the method which simplifies our analysis. This built-in stability at all length scales and the a posteriori error indicator additionally allows for the generation of parameter-specific meshes starting from a common coarse initial mesh. A DPG discretization always produces a symmetric positive definite stiffness matrix. This feature allows us to use the most efficient direct solvers for all of our computations. We use the Camellia finite element software package [3,4] for all of our analysis.

Philipp Knechtges, Marek Behr, Stefanie Elgeti

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.