Subdivision surfaces: a new paradigm for thin-shell finite-element analysis

Abstract

We develop a new paradigm for thin-shell finite-element analysis based on the use of subdivision surfaces for: i) describing the geometry of the shell in its undeformed configuration, and ii) generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the Kirchhoff-Love theory of thin shells. The particular subdivision strategy adopted here is Loop's scheme, with extensions such as required to account for creases and displacement boundary conditions. The displacement fields obtained by subdivision are H 2 and, consequently, have a finite Kirchhoff-Love energy. The resulting finite elements contain three nodes and element integrals are computed by a onepoint quadrature. The displacement field of the shell is interpolated from nodal displacements only. In particular, no nodal rotations are used in the interpolation. The interpolation scheme induced by subdivision is nonlocal, i. e., the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighboring elements. However, the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface. Numerical tests, including the Belytschko et al. [7] obstacle course of benchmark problems, demonstrate the high accuracy and optimal convergence of the method.

An atomistic-based finite deformation membrane for single layer crystalline films

“…For approximation of the deformation, subdivision surfaces based on Loop's scheme are used. This technique originates from the ÿeld of computational geometry and has recently been applied to thin shell analysis (Cirak et al, 2000;Cirak and Ortiz, 2001). In this method, a control mesh of triangular elements with only translational degrees-of-freedom is used to construct a smooth (H 2 ) surface.…”

Structure-preserving model reduction for mechanical systems

“…Wavelet decomposition techniques are naturally hierarchical and multiresolutional, and provide a mathematical framework on which to form adaptive basis methods for finite-element elasticity. In fact, these methods are strongly tied to new subdivision methods for representing the dynamics of thin shells [5]. Note that the model reduction problem for traditional dynamic thin-shell models is more complicated than that for three-dimensional elasticity, due to the underlying geometry of the configuration space and the way essential boundary conditions are treated.…”

“…Note that the model reduction problem for traditional dynamic thin-shell models is more complicated than that for three-dimensional elasticity, due to the underlying geometry of the configuration space and the way essential boundary conditions are treated. The subdivision thin-shell models are an exception to this rule, because their configuration spaces are Euclidean [5]. We do not give a discussion of error bounds in this work; this is an important but separate issue.…”

IGA: A Simplified Introduction and Implementation Details for Finite Element Users

“…Kagan et al [7] introduced a B-spline based FE approach that combines the geometrical design and the analysis systems into a common framework. Later, based on the subdivision of surface scheme Cirak et al [8] presented a different paradigm for describing and modelling of the thin shells geometries in the FEA framework. Based on the motivation of combining the long available CAD systems with the simulation approaches, Hughes et al [3] introduced a new numerical method, popularly known as Isogeometric analysis (IGA).…”