A dual lumping procedure for static condensation in mixed NURBS-based isogeometric elements with optimal convergence rates for arbitrary open knot vectors

Abstract

Locking is a common effect in finite element and isogeometric analysis. In the case of plates, transverse shear locking is most prominent, for shells several other types of locking exist. A common cure are mixed methods that introduce additional fields of unknowns into the variational formulation. These fields reduce constraints and thus alleviate locking significantly. As a drawback, the discretized additional fields increase computational costs significantly. These fields are often eliminated by static condensation, which requires the inverse of a part of the stiffness matrix. In Lagrange-based finite elements, this inverse is computed on element level, due to a discontinuous interpolation of additional fields. Since isogeometric analysis features higher continuity, static condensation must be performed on patch level, which requires a costly matrix inversion on that level. In this contribution, the virtual shear parameters of a mixed isogeometric plate formulation are interpolated by enhanced approximate dual basis functions. This allows to conduct row-sum lumping of the relevant matrix part at a minimal loss of accuracy, since this part becomes diagonal dominant. For a properly chosen integration space, this lumped matrix becomes the identity matrix. Thus, the proposed condensation procedure does not require an inversion anymore. The crucial and novel point is the proposed treatment of knot vectors with limited internal continuity. With the help of several single- and multi-patch examples, both with full and with limited internal continuity, we show that the proposed procedure obtains optimal error convergence rates in all cases, while without these alterations, convergence rates are significantly deteriorated.

Figures (18)

• Figure 1: Comparison of NURBS and their approximate dual basis functions for $p=q=r=s=5$ using the Quadratic plate with complex loading from Sec. \ref{['sec:numerical_exmples']}
• Figure 2: Mapping between element in physical domain $\Omega^e$, element in parametric domain $\tilde{\Omega}^e$ and parent element $\bar{\Omega}^e$
• Figure 3: Sparsity of mixed system matrix for benchmark example \ref{['sec:numerical_exmples']} examined in Sec. \ref{['sec:quadraticPlate_ud']} in case $t=0.1~m$, $r=p=5$ and $40\times40$ elements for all sub-steps of the proposed efficient static condensation procedure.
• Figure 4: Initial meshes with isolines (solid), control points (circles) and control polygon (dashed)
• Figure 5: Initial meshes with patch interface (bold red solid), isolines (solid), control points (circles) and control polygon (dashed)

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5