A novel hybrid isogeometric element based on two-field Hellinger–Reissner principle to alleviate different types of locking

Abstract

In the present work, novel hybrid elements are proposed to alleviate the locking anomaly in non-uniform rational B-spline-based isogeometric analysis (IGA) using a two-field Hellinger–Reissner variational principle. The proposed hybrid elements are derived by adopting the independent interpolation schemes for displacement and stress fields. The key highlight of the present study is the choice and evaluation of higher-order terms for the stress interpolation function to provide a locking-free solution. Furthermore, the present study demonstrates the efficacy of the proposed elements with the treatment of several two-dimensional linear-elastic benchmark problems alongside the conventional single-field IGA, Lagrangian-based finite element analysis (FEA), and hybrid FEA formulation. It is shown that the proposed class of hybrid elements performs effectively for analyzing the nearly incompressible problem domains that are severely affected by volumetric locking along with the thin plate and shell problems where the shear locking is dominant. A better coarse mesh accuracy of the proposed method in comparison with the conventional formulation is demonstrated through various numerical examples. Moreover, the formulation is not restricted to the locking-dominated problem domains but can also be implemented to solve the problems of general form without any special treatment. Thus, the proposed method is robust, most efficient, and highly effective against both shear and volumetric locking.

Appendix I: Geometric data for modeling the base coarse mesh for the presented problems

The section is intended to provide the required control points coordinates and the respective weights to construct the initial mesh for the stated problem domains, see tables 1, 2, 3, 4 and 5. The sequence of the refined meshes are modeled by employing the knot insertion or the degree elevation algorithms in one or both direction. For the reader’s interest, the MATLAB codes based on these algorithms can be found in an open-source code library called NURBS toolbox [56]. The subroutines named nrbkntins and nrbdegelev are of particular interest for the refinement strategies.

Table 1 Control point coordinates ($P_{i,j} = [x \quad y \quad z]$) for modeling the initial coarse mesh (single element) for a rectangular beam of length L and thickness t with linear basis along $\xi$ and $\eta$ direction (weights associated with each control point is 1, knot vector along $\xi$ and $\eta$ direction is $\left[ 0,0,1,1 \right]$).

Table 2 Control point coordinates and respective weights ($P_{i,j} = [x \quad y \quad z \quad w]$) for modeling a single element representing the problem domain of a curved beam with quadratic basis along $\xi$ and linear basis along $\eta$ direction, the respective knot vectors are $\varvec{\Xi }= \left[ 0,0,0,1,1,1 \right]$ and $\varvec{{\mathcal {H}}} = \left[ 0,0,1,1 \right]$.

Table 3 Control point coordinates ($P_{i,j} = [x \quad y \quad z]$) for modeling a single element representing the domain of a Cook’s membrane problem with linear basis along $\xi$ and $\eta$ direction (weights associated with all the control points = 1 and knot vector along $\xi$ and $\eta$ direction is $\left[ 0,0,1,1 \right]$).

Table 4 Control point coordinates ($P_{i,j} = [x \quad y \quad z]$) and respective weights ($w_{i,j}$) for modeling a quarter portion of a plate with a hole problem using two quadratic elements (two elements along $\xi$ direction and one element along $\eta$ direction), knot vectors along $\xi$ and $\eta$ are given as; $\varvec{\Xi }= \left[ 0, 0, 0, 0.5, 1, 1, 1 \right]$, $\varvec{{\mathcal {H}}} = \left[ 0, 0, 0, 1, 1 ,1 \right]$, radius $= r = 1$, length $= L = 4$, weight $=w^{cp}=0.5(1 + 1/\sqrt{2})$.

Table 5 Control point coordinates ($P_{i,j} = [x \quad y \quad z]$) and respective weights ($w_{i,j}$) for modeling a quarter portion of a plate with a hole problem using two cubic elements (two elements along $\xi$ direction and one element along $\eta$ direction), knot vectors along $\xi$ and $\eta$ are given as; $\varvec{\Xi }= \left[ 0,0, 0, 0, 0.5, 1, 1, 1, 1 \right]$, $\varvec{{\mathcal {H}}} = \left[ 0, 0,0,0,1, 1, 1,1 \right]$.