Why Shell Analysis Needs Suitable Discretization Methods
From complex architectural roofs to lightweight engineering structures, thin shells are everywhere in modern design. This blog post explores how advanced isogeometric and scaled-boundary methods can improve the numerical analysis of such structures while enabling flexible and efficient mesh generation.
When developing a Finite Element (FE) approximation method for engineering analysis, it is essential to represent the physical and geometrical properties of structural components as accurately as possible. At the same time, the underlying numerical scheme should require only minimal regularity assumptions. Ideally, the FE spaces are easy to construct and flexible enough to handle a wide range of applications. In other words, one aims for a discretization approach that is truly suitable for analysis in practice.
In the context of shell analysis – the investigation of thin-walled structures – this objective can be achieved particularly well by combining a Reissner-Mindlin (RM) shell model with the discretization paradigm of IsoGeometric Analysis (IGA) [Hughes et al., 2005].
Isogeometric Analysis Meets the Reissner-Mindlin Shell Model
One of the main advantages of IGA is the direct integration of CAD models into the analysis step. Since the finite element spaces are spanned by the same type of functions used in CAD software – namely NURBS (Non-Uniform Rational B-Splines) – geometric errors caused by geometry approximation can be avoided.
In addition, the use of a six-variable Reissner-Mindlin formulation with a drilling degree of freedom [Ibrahimbegović, 1994] provides a flexible description of the shell configuration. In this model, the deformation is represented by the displacement of the reference two-dimensional shell mid-surface together with the orientation of so-called fibers that are initially orthogonal to this surface. Their orientation is encoded by a rotation matrix vector field.
This formulation leads to a discretization method that requires only continuous FE functions, which makes the approach comparatively easy to implement. As a consequence, even shells with stiffeners or non-smooth connections – such as the loaded U-beam shown in Fig. 1(a) – can be treated within this framework.
Adaptive Refinement with Truncated Hierarchical B-Splines
Beyond basic discretization, further characteristics are desirable for an analysis-suitable method. One important aspect is efficiency. In many applications it is advantageous to locally enrich the approximation space by increasing the number of degrees of freedom only where needed.
Truncated hierarchical B-splines (THB-splines) provide an elegant way to achieve such adaptive refinement. They allow the finite element mesh to be modified locally without affecting the entire computational domain, as illustrated in Fig. 1(b).
Meshing Challenges for Complex Geometries
Another important aspect is the generation of the computational mesh itself. For complex geometries—such as the roof structure inspired by the Rolex Learning Center shown in Fig. 2(a)—the underlying CAD objects are often highly processed and trimmed.
Constructing a mesh that exactly respects such geometric features can be challenging, particularly if the discretization relies on specific element types such as quadrilateral or tensor-product patches.
Polytopal Meshes and the Scaled Boundary Isogeometric Approach
A much higher degree of flexibility can be achieved by allowing more general mesh elements. Polytopal meshes, similar to those used in the Virtual Element Method (VEM), can be constructed for instance through Voronoi-type tessellations, as shown in Fig. 2 (c).
An alternative strategy is to use a boundary-based representation of the computational domain via Scaled Boundary sogeometric analysis (SB-IGA). In this approach, the domain is decomposed into star-convex blocks (see Fig. 2(b)). Each block is then described by scaling its boundary towards a chosen scaling center, while the boundary itself is represented exactly by NURBS functions in the spirit of isogeometric analysis.
A Collaborative Research Effort on Polytopal FEM
Questions related to intelligent meshing strategies and the comparison of different discretization approaches in solid mechanics are currently being investigated in a joint collaboration between the RPTU Kaiserslautern-Landau and RWTH Aachen University. The work is embedded in the larger DFG research unit FOR 5492 – Polytope Mesh Generation and Finite Element Analysis Methods for Problems in Solid Mechanics.
The research unit brings together scientists from five universities who study polytopal finite element methods from several complementary perspectives, including numerical mathematics, geometry processing, and computational mechanics.
In addition to developing new approaches to mesh generation (PIs: Kobbelt, RWTH Aachen University, and Jüttler, JKU Linz), the project also focuses on computational methods in civil engineering and applied mechanics (PIs: Klinkel, RWTH Aachen University, and Holthusen, FAU Erlangen–Nürnberg). Other research topics include fracture processes in brittle polycrystalline materials (PI: Birk, University of Duisburg–Essen) and the use of machine learning for discretizing scaled-boundary blocks (PI: Chasapi, RWTH Aachen University). Numerical analysis and local refinement strategies constitute another central research direction (PI: Simeon, RPTU Kaiserslautern-Landau).
Exchange and Progress within the Research Network
At the most recent project meeting in Hinterstoder, Austria (see Fig. 3), the partners discussed recent achievements and outlined the next steps of the collaboration.
More about the DFG research group FOR 5492 – Polytope Mesh Generation and Finite Element Analysis Methods for Problems in Solid Mechanics.
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.
