Combining isogeometric analysis (IGA), finite element methods (FEM) and embedded mesh (EM) coupling for contact problems
Final Report Abstract
The modeling and numerical simulation of contact problems and the physical phenomena associated with them (e.g. friction, adhesion) represent a challenge in many areas of civil engineering. In recent years, many advances have been made in the numerical simulation of such problems, with a special focus on the development of formulations using the finite element method (FEM). Such formulations rely on an accurate representation of the contact surface, which can be partially achieved by using a fine mesh resolution or using higher-order elements. However, with a discretization based on Lagrange polynomials as shape functions, the resulting mesh doesn’t deliver a smooth representation of curved surfaces and is limited to C0 continuity at inter-element boundaries. In other words, the discretized surface has a non-continuous field of normal vectors, which introduces additional complexity to contact algorithms. To overcome these drawbacks, approaches based on isogemetric analysis (IGA) have been developed. In IGA, the discretization is based on the exact CAD geometry, which employs smooth and high-order functions, such as non-uniform rational B-splines (NURBS). Such shape functions provide a higher and tailored inter-element continuity with an accurate representation of the contact surface. However, the generation of isogeometric meshes can be challenging for complex geometries. Moreover, the propagation of the higher-order continuity into the bulk volume of the domain might not be needed for non-smooth problem classes due to their inherently reduced regularity, such as unilateral contact. Looking at both FEM and IGA formulations, both approaches offer advantages that can be combined to deliver new possibilities for the solution of contact problems. Therefore, the goal of this project was to combine the advantages of IGA and FEM, distinguishing between the discretization of the body’s contact interface and its interior. The contact interface, represented by a NURBS boundary representation, is extruded, creating a NURBS boundary layer mesh. Furthermore, the inner bulk volume is discretized with a regular FEM or IGA hexahedral mesh in the reference configuration. The resulting overlapping meshes are coupled by using appropriate embedded mesh methods. During the funding period, the first step was to establish a pre-processing framework for the in-house research code 4C. This mainly consists of two parts: the generation of an isogeometric boundary layer and the creation of an enclosed Cartesian mesh. Three methods were investigated for offsetting curves and surfaces represented by NURBS shape functions. In terms of numerical methods, an approach for embedded/overlapping mesh coupling framework was developed in 4C. Specifically, a Mortar/Lagrange Multiplier embedded mesh framework was implemented and investigated. For the integration of cut elements, a previously existing 3-dimensional tessellation algorithm was expanded to be compatible with NURBS-based interfaces. Although the investigation of Nitsche Methods and other stabilization methods could not be achieved in the duration of the project, the results obtained provide a solid foundation for expanding the embedded mesh framework to include such approaches in the future.
Publications
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Coupling of NURBS boundary layer meshes with bulk finite element meshes using mortar methods, 10th Contact Mechanics International Symposium (CMIS 2022), Chexbres, Switzerland, May 23 - 25, 2022, & FE ohne Schnee, (2022), Mittelberg, Austria, July 16 - 19, 2022
E.G. Loera Villeda & A. Popp
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Towards an embedded mesh approach for treating overlapping isogeometric and finite element meshes, 9th GACM Colloquium on Computational Mechanics (GACM 2022), Essen, Germany, September 21 - 23, 2022
E.G. Loera Villeda & A. Popp
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Towards an embedded mesh approach for isogeometric boundary layers in contact mechanics, 7th International Conference on Computational Contact Mechanics (ICCCM 2023), Torino, Italy, July 5 - 7, 2023
E.G. Loera Villeda, I. Steinbrecher & A. Popp
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Towards an embedded mesh approach for overlapping isogeometric and finite element meshes in contact mechanics, 11th International Conference on Isogeometric Analysis (IGA 2023), Lyon, France, June 18 - 21, 2023
E.G. Loera Villeda, I. Steinbrecher & A. Popp
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An embedded mesh approach for coupling tailored isogeometric and finite element meshes for contact problems, 9th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2023), Lisbon, Portugal, June 3 - 7, 2024
E.G. Loera Villeda, I. Steinbrecher & A. Popp
