A mixed isogeometric plane stress and plane strain formulation with different continuities for the alleviation of locking

[EN] Isogeometric analysis was founded by Hughes et al. and tries to unify computer aided design (CAD) and finite element analysis (FEA) by using the same model for geometry representation and analysis. Therefore, non-uniform rational B-splines (NURBS) and other kinds of splines are used as shape functions of the finite elements. Due to the exact representation of the geometry, analysis results can be improved. Furthermore, many fast and numerically stable algorithms have been developed that exhibit favourable mathematical properties.In mixed formulations stresses and/or strains or pressures are approximated independently and in addition to the usual displacement approximation. Using such methods is more robust and offers more accurate results. Hence, mixed formulations are employed to solve incompressible elasticity problems for instance.Recent investigations have already combined isogeometric analysis and mixed formulations in order to benefit from the advantages of both methods.In this contribution, a mixed isogeometric method is proposed in order to improve the analysis results and to counteract locking. Therefore, spline basis functions are used and the displacement shape functions of a two-dimensional isogeometric plane stress and plane strain element are supplemented by independent stress shape functions. These additional stress shape functions are chosen to be of one order lower compared to the displacement shape functions, but with adapted continuity.Evaluating the error norms for several examples, it is shown that the proposed mixed method leads to an improved accuracy of results compared to a standard isogeometric formulation and is able to counteract locking. Furthermore, the influence of the continuity of the stress shape functions is shown.Stammen, L.; Dornisch, W. (2022). A mixed isogeometric plane stress and plane strain formulation with different continuities for the alleviation of locking. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 109-118. https://doi.org/10.4995/YIC2021.2021.12554OCS10911

An Isogeometric Element Formulation for Linear Two-Dimensional Elasticity Based on the Airy Equation

[EN] 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.Held, S.; Dornisch, W.; Azizi, N. (2022). An Isogeometric Element Formulation for Linear Two-Dimensional Elasticity Based on the Airy Equation. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 129-138. https://doi.org/10.4995/YIC2021.2021.12598OCS12913

An isogeometric mortar method for the coupling of multiple NURBS domains with optimal convergence rates

We investigate the mortar finite element method for second order elliptic boundary value problems on domains which are decomposed into patches Ω_k with tensor-product NURBS parameterizations. We follow the methodology of IsoGeometric Analysis (IGA) and choose discrete spaces X_h,k on each patch Ω_k as tensor-product NURBS spaces of the same or higher degree as given by the parameterization. Our work is an extension of [12] and highlights several aspects which did not receive full attention before. In particular, by choosing appropriate spaces of polynomial splines as Lagrange multipliers, we obtain a uniform infsup-inequality. Moreover, we provide a new additional condition on the discrete spaces X_h,k which is required for obtaining optimal convergence rates of the mortar method. Our numerical examples demonstrate that the optimal rate is lost if this condition is neglected

Методы сравнения качества наборов дефектоскопических материалов для капиллярного контроля

An isogeometric Reissner-Mindlin shell derived from the continuum theory is presented. The geometry is described by NURBS surfaces. The kinematic description of the employed shell theory requires the interpolation of the director vector and of a local basis system. Hence, the definition of nodal basis systems at the control points is necessary for the proposed formulation. The control points are in general not located on the shell reference surface and thus, several choices for the nodal values are possible. The proposed new method uses the higher continuity of the geometrical description to calculate nodal basis system and director vectors which lead to geometrical exact interpolated values thereof. Thus, the initial director vector coincides with the normal vector even for the coarsest mesh. In addition to that a more accurate interpolation of the current director and its variation is proposed. Instead of the interpolation of nodal director vectors the new approach interpolates nodal rotations. Account is taken for the discrepancy between interpolated basis systems and the individual nodal basis systems with an additional transformation. The exact evaluation of the initial director vector along with the interpolation of the nodal rotations lead to a shell formulation which yields precise results even for coarse meshes. The convergence behavior is shown to be correct for k-refinement allowing the use of coarse meshes with high orders of NURBS basis functions. This is potentially advantageous for applications with high numerical effort per integration point. The geometrically nonlinear formulation accounts for large rotations. The consistent tangent matrix is derived. Various standard benchmark examples show the superior accuracy of the presented shell formulation. A new benchmark designed to test the convergence behavior for free form surfaces is presented. Despite the higher numerical effort per integration point the improved accuracy yields considerable savings in computation cost for a predefined error bound

Treatment of Reissner–Mindlin shells with kinks without the need for drilling rotation stabilization in an isogeometric framework

This work presents a framework for the computation of complex geometries containing intersections of multiple patches with Reissner-Mindlin shell elements. The main objective is to provide an isogeometric finite element implementation which neither requires drilling rotation stabilization, nor user interaction to quantify the number of rotational degrees of freedom for every node. For this purpose, the following set of methods is presented. Control points with corresponding physical location are assigned to one common node for the finite element solution. A nodal basis system in every control point is defined, which ensures an exact interpolation of the director vector throughout the whole domain. A distinction criterion for the automatic quantification of rotational degrees of freedom for every node is presented. An isogeometric Reissner-Mindlin shell formulation is enhanced to handle geometries with kinks and allowing for arbitrary intersections of patches. The parametrization of adjacent patches along the interface has to be conforming. The shell formulation is derived from the continuum theory and uses a rotational update scheme for the current director vector. The nonlinear kinematic allows the computation of large deformations and large rotations. Two concepts for the description of rotations are presented. The first one uses an interpolation which is commonly used in standard Lagrange-based shell element formulations. The second scheme uses a more elaborate concept proposed by the authors in prior work, which increases the accuracy for arbitrary curved geometries. Numerical examples show the high accuracy and robustness of both concepts. The applicability of the proposed framework is demonstrated