- Integration of design (CAD) and analysis (FEM)
- Finite Elemente Method with NURBS shape functions of high continuity
- Exploit the higher continuity for the development of innovative shell elements
- Simple representation of large rotations for shear deformable shell formulations
- Intrinsical avoidance of geometrical locking effects
Isogeometric Analysis (IGA) received a great deal of attention in the past decade and presents a suitable numerical method to solve a number of problems of academic and industrial interest. There are many benefits coming along with smooth function spaces (as NURBS) and higher inter-element continuity. In particular for shell finite elements significant benfits come along with a smoother basis like NURBS. For instance, a NURBS discretization of higher continuity enables a pointwise unique description of the director field within a whole patch.
Hierarchic, shear deformable shell formulations
The higher continuity of isogeometric discretizations provoked a renaissance of classical shell models neglecting transverse shear deformation, for instance Kirchhoff-Love, because corresponding requirements for the smoothness of ansatz spaces (min. C1 within a patch) are easily met. However, shear deformable shell formulations with Reissner-Mindlin kinematics and threedimensional shell models enable the use of C0-continuous ansatz spaces. Nevertheless, the higher continuity can be exploited to develop innovative shear deformable shell elements.
Hierarchic shell formulations of Reissner-Mindlin type are developed through hierarchic enrichment of the Kirchhoff-Love model. Via hierarchic reparametrization of the kinematic equations transverse shear locking can be a priori avoided within a pure primal method. The concept is equally applied for shear deformable beam, plate and shell formulations, whereas two hierarchic parameterizations can be distinguished.
The parametrization of hierarchic rotations (RM-hr) as well as the parametrization of hierarchic displacements (RM-hd) results in formulations being intrinsically free from transverse shear locking.
Simple representation of large rotations for shear deformable shell formulations
A geometrically nonlinear analysis with shear deformable shell formulations requires a special treatment and a multiplicative decomposition of large rotations. Via elegant reparametrization in hierarchic shell formulations, also in this context an innovative concept can be developed. The underlying assumption is that transverse shear rotations are rather small for typical shell problems, whereas total rotations may be arbitrarily large. This assumption results in a additive decomposition of the Green-Lagrange strains into fully nonlinear parts of the roation-free Kirchhoff-Love type and hierarchically added, linearized transverse shear parts. Thus the total rotations are additively decomposable and can be represented in a simple fashion.
Materially non-linear analyses
Assuming linearized transverse shear poses the question of possible limitations regarding simulations of not only large rotations but large strains. Especially elasto-plastic material laws that use the deviatoric part of the strains to determine the actual stress state are of interest. To model non-linear constitutive laws, the C++ open source library MUESLI of IMDEA Materials Institute from Madrid is linked to the institute’s shell code.
Higher inter-element continuities of isogeometric ansatz spaces may also have disadvantages with respect to quality of results. In particular membrane locking proved to be extremely dominant. For maximum continuity the occurring oscillations in membrane forces do not vanish with higher orders and lead for primal isogeometric shell elements under certain circumstances to very bad results.
Classical concepts known from finite element technology are usually not directly transferable to isogeometric finite elements because the higher inter-element continuity prohibits elementwise treatment. With the Mixed-Dipslacement (MD) concept a variational method has been developed in order to efficiently avoid geometric locking effects. The combination of hierarchic shell formulations based on hierarchic displacements and the MD concept to avoid membrane locking leads to fully locking free shear deformable shell formulations (RM-hd-MD). The snap-through benchmark problem of a cylindrical segment under large rotations displays the superior behavior of the presented RM-hd-MD formulation compared to a primal Kirchhoff-Love ( KL) formulation. Independent of the polynomial order, with the RM-hd-MD formulation smooth results of high quality can be obtained for the membrane force. The primal KL elements allow only a bad approximation of the expected solution and show strong oscillations in the results for the membrane force.
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