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An isogeometric element formulation for linear two-dimensional elasticity based on the Airy equation
(2022)
The aim of this work is to derive a formulation for linear two-dimensional elasticity using just one degree of freedom. This degree of freedom is used to directly discretize the Airy bipotential equation, which requires higher order basis functions. Isogeometric structural analysis is based on shape functions of the geometry description in Computer-Aided design software.
These shape functions can easily fulfill the continuity requirement of the bipotential equation.
Thus, an Airy element formulation can be obtained through isogeometric methods. In this contribution Non-Uniform Rational B-splines are used to discretize the domain and to solve the occurring differential equations. Numerical examples demonstrate the accuracy of the evolved formulation for a quadratic plate under different load situations.
A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition
(2024)
A curved non-isoparametric Reissner–Mindlin shell element is developed for analyzing thin-walled structures. The standard kinematic description of the element requires the calculation of the director vector. To address this demand accurately, similar to isogeometric analysis (IGA), the geometry is defined by utilization of the non-uniform rational B-splines (NURBS) imported directly from computer-aided design (CAD) files. Then, shape functions of the Legendre spectral element method (SEM) are used to interpolate the displacements. Consequently, the shell director vector and Jacobian of the transformation are calculated properly according to the presented formulation. On the other hand, in Legendre SEM combined with Gauss–Lobatto–Legendre quadrature, the integration points and the element nodes coincide. Thus, the easily computable local coordinate systems at the integration points can be used directly as nodal basis systems. A separate calculation of nodal basis systems at control points, which is the source of either complexity or error in IGA shells, is not required. Given the condition number of the stiffness matrix in the developed method, super high-order elements can also be used. Very high order p-refined elements are used in addition to h-refinement of the mesh to show the capability of higher order elements to analyze problems without mesh refinement. The validity and convergence rate of the method are investigated and verified through various cases of h- and p-refinement in challenging obstacle course problems.