This paper proposes a shape optimization algorithm based on the principles of Isogeometric Analysis (IGA) in which the parameterization of the geometry enters the problem formulation as an additional PDE-constraint. Inspired by the isoparametric principle of IGA, the parameterization and the governing state equation are treated using the same numerical technique. This leads to a scheme that is comparatively easy to differentiate, allowing for a fully symbolic derivation of the gradient and subsequent gradient-based optimization. To improve the efficiency and robustness of the scheme, the basis is re-selected during each optimization iteration and adjusted to the current needs. The scheme is validated in two test cases.

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The first step towards applying isogeometric analysis techniques to solve PDE problems on a given domain consists in generating an analysis-suitable mapping operator between parametric and physical domains with one or several patches from no more than a description of the boundary contours of the physical domain. A subclass of the multitude of the available parameterization algorithms are those based on the principles of Elliptic Grid Generation (EGG) which, in their most basic form, attempt to approximate a mapping operator whose inverse is composed of harmonic functions. The main challenge lies in finding a formulation of the problem that is suitable for a computational approach and a common strategy is to approximate the mapping operator by means of solving a PDE-problem. PDE-based EGG is well-established in classical meshing and first generalization attempts to spline-based descriptions (as is mandatory in IgA) have been made. Unfortunately, all of the practically viable PDE-based approaches impose certain requirements on the employed spline-basis, in particular global C≥1-continuity.

This paper discusses an EGG-algorithm for the generation of planar parameterizations with locally reduced smoothness (i.e., with support for locally only C0-continuous bases). A major use case of the proposed algorithm is that of multipatch parameterizations, made possible by the support of C0-continuities. This paper proposes a specially-taylored solution algorithm that exploits many characteristics of the PDE-problem and is suitable for large-scale applications. It is discussed for the single-patch case before generalizing its concepts to multipatch settings. This paper is concluded with three numerical experiments and a discussion of the results.@en

The construction of volumetric parametrizations for computational domains is a key step in the pipeline of isogeometric analysis. Here, we investigate a solution to this problem based on the mesh deformation approach. The desired domain is modeled as a deformed configuration of an initial simple geometry. Assuming that the parametrization of the initial domain is bijective and that it is possible to find a locally invertible displacement field, the method yields a bijective parametrization of the target domain. We compute the displacement field by solving the equations of nonlinear elasticity with the neo-Hookean material law, and we show an efficient variation of the incremental loading algorithm tuned specifically to this application. In order to construct the initial domain, we simplify the target domain's boundary by means of an L²-projection onto a coarse basis and then apply the Coons patch approach. The proposed methodology is not restricted to a single patch scenario but can be utilized to construct multi-patch parametrizations with naturally looking boundaries between neighboring patches. We illustrate its performance and compare the result to other established parametrization approaches on a range of two-dimensional and three-dimensional examples.

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The aim of this dissertation is to introduce the concept of PDE-based parameterization using Isogeometric Analysis (IGA) techniques and to present first results. IGA is a numerical technique that employs the spline- and NURBS-based geometric modeling tools of Computer Aided Geometric Design (CAGD) as a basis for numerical simulation using the principles of Finite Element Analysis (FEA). Thiswork proposes techniques that employ a parametric domain comprised of one (singlepatch) or several (multipatch) unit quadrilaterals and find an analysis-suitable mapping operator that parameterizes the physical domain. Here, the only input is a NURBS-based correspondence between the boundaries of the parametric and physical domain. To achieve this, this work adopts the principles of Elliptic Grid Generation (EGG), a PDE-based technique from classical meshing, and proposes discretizations that are suitable for an IGA-based computational workflow. The PDE-based problem formulation is motivated by the prospect of employing the same numerical technique for the geometrical as well as the computational aspects of the numerical simulation pipeline. @en

Isogeometric Analysis (IgA) has become an accepted framework for the mod-elling, simulation and optimization (MSO) of engineering processes. However, the fully automatized generation of analysis-suitable parameterizations of geometries as they arise in practical workflows is still a challenging task, which often requires application-specific parameterization approaches. In this article we present a practical approach [6] based on the principles of Elliptic Grid Generation (EGG) for the efficient on-demand generation of analysis-suitable spline-based parameterizations. Starting from a (point cloud) description of the boundary provided by the existing MSO-pipeline, an inverse nonlinear Poisson-type problem is solved to obtain a folding-free (planar) parameterization of the entire domain. The non-linearity is efficiently treated with a globalized hierarchical Newton approach. Automatized boundary contour reparameterization techniques are employed to improve the parametric properties from a numerical viewpoint, such as orthogonal isolines and equally-sized cells. The use of curved instead of straight-sided elements allows us to arrive at an accurate description of the target domain with fewer elements and thus potentially lower computational effort. Numerical experiments with screw-compressor geometries demonstrate that the proposed algorithm reliably produces high-quality parameterizations typically within 3−4 Newton-iterations, even in the presence of extreme aspect-ratios. This makes it particularly attractive for the on-demand application within an automatized industrial MSO-pipeline. To support the demands of modern high-performance computing hardware, only a moderate number of sufficiently large and structured patches is generated which can be mapped one-by-one to the different devices (CPUs/GPUs) with only little communication overhead. Topology changes are avoided in time-dependent and shape-optimization settings. Finally, the spline-based geometry description can be transformed into a classical mesh by performing a large number of function evaluations in the mapping operator. Its continuous nature allows for feature-based (structured and unstructured) refinement and arbitrary element densities without increasing the complexity of the meshing process.

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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.

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The fully automated generation of computational meshes for twin-screw machine geometries constitutes a mandatory aspect for the numerical simulation (and shape-optimization) of these devices but proves to be a challenging task in practice. Therefore, the successful generation of computational meshes requires sophisticated mathematical tools. Commercially available classical mesh generators can produce high quality meshes from no more than a description of the rotor contours. However, since we are particularly interested in numerical simulations using the principles of Isogeometric Analysis, a spline-based geometry description rather than a classical mesh is needed.
In this paper, we propose a practical approach for the automated generation of spline-based twin-screw machine geometry parameterizations in two dimensions. For this purpose, we adopt the principles of Elliptic Grid Generation and present a parameterization algorithm that is compatible with an automated simulation pipeline based on the principles of isogeometric analysis.
To demonstrate the proposed techniques, we apply them to an example geometry and present the resulting parameterizations. Finally, we give a qualitative explanation of how the discussed techniques can be utilized to generate geometry parameterizations in three dimensions and their applications to shape-optimization on a variable rotor-pitch.@en

The generation of an analysis-suitable computational grid from a description of no more than its boundaries is a common problem in numerical analysis. Most classical meshing techniques for finite-volume, finite-difference or finite-element applications such as the Advancing Front Method (Schöberl, 1997), Delaunay Triangulation (Triangle, 1996) and elliptic or hyperbolic meshing schemes (Thompson et al., 1998) operate with linear or multi-linear but straight-sided elements for the generation of structured and unstructured meshes, respectively, whereas the generation of high-quality curved meshes is still considered a major challenge. A recent development is the introduction of Isogeometric Analysis (IgA) (Hughes et al., 2005), which can be considered as a natural high-order generalisation of the finite-element method. A description of the geometry Ω¯ is accomplished via a mapping operator x:Ωˆ→Ω that maps the unit hypercube in Rⁿ onto an approximation Ω of Ω¯ utilizing a linear combination of higher-order spline functions. The numerical simulation is then carried out in the computational domain Ωˆ via a ‘pull back’ using the mapping operator x. The advantage is that the flexibility of higher-order spline-functions usually allows for an accurate description of Ω¯ with much fewer elements which can significantly reduce the computational effort required for this step compared to traditional low-order methods. Furthermore, an analytical description of the geometry can be turned back into a traditional (structured or unstructured) grid by performing a large number of function evaluations in x. This can, for instance, be utilized for local refinement without the need for remeshing. A potential drawback of curved instead of linear elements is that the meshing techniques required for the creation of folding-free mappings tend to be more sophisticated and that it is a less trivial task to verify that the resulting mapping is indeed bijective. For the purpose of creating folding-free mappings utilizing spline-functions, we will present an algorithm adopting the principles of Elliptic Grid Generation (EGG) whose basic principles have been adapted to meet the needs of IgA. In R², EGG has particularly appealing properties since bijectivity of the resulting mapping is guaranteed as long as the numerical accuracy to the computational approach of the mapping is sufficient. We will present an algorithm that is capable of generating folding-free mappings from a large number of geometry contours, including complicated geometries from industrial applications with extreme aspect ratios. This is accomplished by combining EGG with automatized reparameterization techniques and a sophisticated numerical approach for solving resulting governing (nonlinear) equations. The algorithm is equipped with the means to verify the bijectivity of the resulting mapping and with automatized defect-correction methods in case a violation of bijectivity is detected. Furthermore, we will present possible strategies for the generation of folding-free mappings for certain types of volumetric geometries by combining EGG with transfinite-interpolation as well as a number of other applications such as time-dependent settings. All applications are provided with example geometries.

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