TY  - THES
A1  - Malik, Irfan
T1  - An adaptive contact formulation for Isogeometric Finite Element Analysis
N2  - Numerical simulation of physical phenomena, like electro-magnetics, structural and fluid mechanics is essential for the cost- and time-efficient development of mechanical products at high quality. It allows to investigate the behavior of a product or a system far before the first prototype of a product is manufactured.
This thesis addresses the simulation of contact mechanics. Mechanical contacts appear in nearly every product of mechanical engineering. Gearboxes, roller bearings, valves and pumps are only some examples. Simulating these systems not only for the maximal/minimal stresses and strains but for the stress-distribution in case of tribo-contacts is a challenging task from a numerical point of view.
Classical procedures like the Finite Element Method suffer from the nonsmooth representation of contact surfaces with discrete Lagrange elements. On the one hand, an error due to the approximate description of the surface is introduced. On the other hand it is difficult to attain a robust contact search because surface normals can not be described in a unique form at element edges.
This thesis introduces therefore a novel approach, the adaptive isogeometric contact formulation based on polynomial Splines over hierarchical T-meshes (PHT-Splines), for the approximate solution of the non-linear contact problem. It provides a more accurate, robust and efficient solution compared to conventional methods. During the development of this method the focus was laid on the solution of static contact problems without friction in 2D and 3D in which the structures undergo small deformations.
The mathematical description of the problem entails a system of partial differential equations and boundary conditions which model the linear elastic behaviour of continua. Additionally, it comprises side conditions, the Karush-Kuhn-Tuckerconditions, to prevent the contacting structures from non-physical penetration. The mathematical model must be transformed into its integral form for approximation of the solution. Employing a penalty method, contact constraints are incorporated by adding the resulting equations in weak form to the overall set of equations. For an efficient space discretization of the bulk and especially the contact boundary of the structures, the principle of Isogeometric Analysis (IGA) is applied. Isogeometric Finite Element Methods provide several advantages over conventional Finite Element discretization. Surface approximation with Non-Uniform Rational B-Splines (NURBS) allow a robust numerical solution of the contact problem with high accuracy in terms of an exact geometry description including the surface smoothness.
The numerical evaluation of the contact integral is challenging due to generally non-conforming meshes of the contacting structures. In this work the highly accurate Mortar Method is applied in the isogeometric setting for the evaluation of contact contributions. This leads to an algebraic system of equations that is linearized and solved in sequential steps. This procedure is known as the Newton Raphson Method. Based on numerical examples, the advantages of the isogeometric approach
with classical refinement strategies, like the p- and h-refinement, are shown and the influence of relevant algorithmic parameters on the approximate solution of the contact problem is verified. One drawback of the Spline approximations of stresses though is that they lack accuracy at the contact edge where the structures change their boundary from contact to no contact and where the solution features a kink. The approximation with smooth Spline functions yields numerical artefacts in the form of non-physical oscillations.
This property of the numerical solution is not only a drawback for the
simulation of e.g. tribological contacts, it also influences the convergence properties of iterative solution procedures negatively. Hence, the NURBS discretized geometries are transformed to Polynomial Splines over Hierarchical T-meshes (PHT-Splines), for the local refinement along contact edges to reduce the artefact of pressure oscillations. NURBS have a tensor product structure which does not allow to refine only certain parts of the geometrical domain while leaving other parts unchanged. Due to the Bézier Extraction, lying behind the transformation from NURBS to PHT-Splines, the connected mesh structure is broken up into separate elements. This allows an efficient local refinement along the contact edge.
Before single elements are refined in a hierarchical form with cross-insertion, existing basis functions must be modified or eliminated. This process of truncation assures local and global linear independence of the refined basis which is needed for a unique approximate solution. The contact boundary is a priori unknown. Local refinement along the contact edge, especially for 3D problems, is for this reason not straight forward. In this work the use of an a posteriori error estimation procedure, the Super Convergent Recovery Solution Based Error Estimation Scheme, together with the Dörfler Marking Method is suggested for the spatial search of the contact edge.
Numerical examples show that the developed method improves the quality of solutions along the contact edge significantly compared to NURBS based approximate solutions. Also, the error in maximum contact pressures, which correlates with the pressure artefacts, is minimized by the adaptive local refinement.
In a final step the practicability of the developed solution algorithm is verified by an industrial application: The highly loaded mechanical contact between roller and cam in the drive train of a high-pressure fuel pump is considered.
KW  - Isogeometrische Analyse
KW  - Elementanalyse
KW  - Isogeometric Analysis
KW  - Error estimation
KW  - adaptive contact
KW  - Finite Element Analysis
KW  - Computational contact mechanics
Y1  - 2022
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20220324-46129
ER  - 
TY  - THES
A1  - Chan, Chiu Ling
T1  - Smooth representation of thin shells and volume structures for isogeometric analysis
N2  - The purpose of this study is to develop self-contained methods for obtaining smooth meshes which are compatible with isogeometric analysis (IGA). The study contains three main parts. We start by developing a better understanding of shapes and splines through the study of an image-related problem. Then we proceed towards obtaining smooth volumetric meshes of the given voxel-based images. Finally, we treat the smoothness issue on the multi-patch domains with C1 coupling. Following are the highlights of each part.

First, we present a B-spline convolution method for boundary representation of voxel-based images. We adopt the filtering technique to compute the B-spline coefficients and gradients of the images effectively. We then implement the B-spline convolution for developing a non-rigid images registration method. The proposed method is in some sense of “isoparametric”, for which all the computation is done within the B-splines framework. Particularly, updating the images by using B-spline composition promote smooth transformation map between the images. We show the possible medical applications of our method by applying it for registration of brain images.

Secondly, we develop a self-contained volumetric parametrization method based on the B-splines boundary representation. We aim to convert a given voxel-based data to a matching C1 representation with hierarchical cubic splines. The concept of the osculating circle is employed to enhance the geometric approximation, where it is done by a single template and linear transformations (scaling, translations, and rotations) without the need for solving an optimization problem. Moreover, we use the Laplacian smoothing and refinement techniques to avoid irregular meshes and to improve mesh quality. We show with several examples that the method is capable of handling complex 2D and 3D configurations. In particular, we parametrize the 3D Stanford bunny which contains irregular shapes and voids.

Finally, we propose the B´ezier ordinates approach and splines approach for C1 coupling. In the first approach, the new basis functions are defined in terms of the B´ezier Bernstein polynomials. For the second approach, the new basis is defined as a linear combination of C0 basis functions. The methods are not limited to planar or bilinear mappings. They allow the modeling of solutions to fourth order partial differential equations (PDEs) on complex geometric domains, provided that the given patches are G1
continuous. Both methods have their advantages. In particular, the B´ezier approach offer more degree of freedoms, while the spline approach is more computationally efficient. In addition, we proposed partial degree elevation to overcome the C1-locking issue caused by the over constraining of the solution space. We demonstrate the potential of the resulting C1 basis functions for application in IGA which involve fourth order PDEs such as those appearing in Kirchhoff-Love shell models, Cahn-Hilliard phase field application, and biharmonic problems.
T3  - ISM-Bericht // Institut für Strukturmechanik, Bauhaus-Universität Weimar - 2020,2 
KW  - Modellierung
KW  - Isogeometrische Analyse
KW  - NURBS
KW  - Geometric Modeling
KW  - Isogeometric Analysis
KW  - NURBS
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20200812-42083
ER  -