When delving into numerical analysis, particularly in the realm of Finite Elements (FEs), one encounters several difficulties and hurdles associated with the central step of discretizing the underlying problem. Various challenges arise, such as incorporating the respective model’s built-in regularity requirements and considering boundary and initial conditions. Not to mention the necessity for a suitable weak formulation that guides to the discrete formulation. Even a step prior, describing the relevant computational domains often demands substantial effort. Despite the advancement of Isogeometric Analysis (IGA) over the past 20 years, bridging the gap between Computer-Aided-Design (CAD) and the concept of Finite Element Methods (FEMs), generating meshes for topologically complex domains remains a hard task. This is partly due to the significant influence that the choice of mesh has on approximation properties.
DFG Project Combines Isogeometric Analysis and the Scaled Boundary Finite Element Method
Hence, within the framework of a two-phase interdisciplinary DFG (Deutsche Forschungsgemeinschaft, German Research Foundation) research project, which involved collaboration between the research groups of Prof. Bernd Simeon (University of Kaiserslautern-Landau (RPTU), applied mathematics group) and Prof. Sven Klinkel (RWTH Aachen, civil engineering group), the goal was to develop a robust numerical method that combines IGA and the Scaled Boundary Finite Element Method (SB-FEM) to model complex geometries in linear and nonlinear solid mechanics. SB-FEM relies on the Boundary Representation Modeling Technique (BRMT), where the body is defined by its boundary using appropriate curves and/or surfaces.
Additionally, a scaling center is chosen to describe the complete body through a radial scaling of the boundary curves or surfaces. In the first phase of the project, the Scaled Boundary Isogeometric Analysis (SB-IGA) was thoroughly investigated. In view of CAD, NURBS (Non-Uniform Rational B-Splines) were used for the boundary description of the computational domains. Furthermore, contrary to classical SB-FEM, the definition of FE spaces on the boundary and in the radial direction was also based on NURBS, following the paradigm of IGA; see Figure 1 a). It was demonstrated how SB-IGA compares to classical Galerkin-based FEMs, with emphasis on stability and convergence studies. Moreover, it was shown that the developed methods are applicable to a wide range of problems, especially in solid mechanics, including nonlinear material laws.
Although initially there seems to be an intrinsic limitation of SB-FEM to star-convex domains, within the project framework, the decomposition of complex domains into starshaped blocks, for instance, via Quadtree, was theoretically explained and numerically tested. Each of these blocks is separately parametrized after choosing a scaling center; compare Fig. 1 (b)-(c). Following a suitable coupling of the different blocks, the concept of SB-IGA can be generalized to complicated geometric structures.
Second Project Phase: Transition to Multiple Scaling Centers and Star-convex Blocks
This transition to multiple scaling centers and star-convex blocks also represented a significant aspect in the second phase of the project. More precisely, in the second part, special requirements for Scaled Boundary (SB) parametrization were examined to obtain globally smooth FE basis functions. Special attention was given to planar C1-regularity across patch and block boundaries. The question of smoothness of the basis functions is crucial, especially when applying SB-IGA to high-order PDE problems. Particularly for trimmed domains, where parts are derived by cutting away or truncating from a given domain, defining C1-continuous FE spaces is non-trivial. Here, the strength of the SB-IGA approach is evident. While the singular scaling center requires special treatment, the SB parametrizations meet the properties of a so-called Analysis-Suitable G1 Parametrization (Collin et al., 2016), enabling implementation of C1 coupling without C1-locking effects.
Furthermore, the boundary description of the domains allows for the exact incorporation of trimming via modified boundary curves. The applicability of the trimmed and C1-coupled SB parametrizations was demonstrated in the context of shells and plates within the Kirchhoff theory, involving complex mesh structures as depicted in Figure 2 below. In addition to addressing questions of higher regularity, an analysis framework was created for 3D solids in boundary representation for problems in nonlinear structural dynamics in the second project phase, too.
With this project, it has been demonstrated that the SB-IGA approach represents a user-friendly method for the construction of FE discretization, which is applicable, at least in the planar case, to arbitrarily complex domains. It is worth mentioning that the block-wise SBIGA approach opens the door to moving beyond classical triangulations based on simplices or quadrilaterals. This is because the appearing star-convex blocks can be defined as new macroelements, allowing the meshes treated in the project to cover polytopal elements in a certain sense. For this reason, the preliminary work will also be utilized in a new, larger DFG research project titled »Polytope Mesh Generation and Finite Element Analysis Methods for Problems in Solid Mechanics« (led by Prof. Sven Klinkel, RWTH Aachen; see here). The latter project will also address other pertinent questions in the realm of SB-IGA, such as how to efficiently locally refine SB meshes and, based on that, how to develop adaptive refinement methods in the context of SB-IGA.
Authors:
Jeremias Arf of the University of Kaiserslautern-Landau (Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau RPTU), Department of Mathematics, Differential-Algebraic Systems Group and Mathias Reichle of the RWTH Aachen University, Chair and Institute of Structural Analysis and Dynamics