TY - GEN A1 - Held, Susanne A1 - Eisenträger, Sascha A1 - Dornisch, Wolfgang T1 - An efficient mass lumping scheme for isogeometric analysis based on approximate dual basis functions T2 - Technische Mechanik N2 - In this contribution, we propose a new mass lumping scheme for explicit dynamics in isogeometric analysis (IGA). To this end, an element formulation based on the idea of dual functionals is developed. Non-Uniform Rational B-splines (NURBS) are applied as shape functions and their corresponding dual basis functions are applied as test functions in the variational form, where two kinds of dual basis functions are compared. The first type are approximate dual basis functions (AD) with varying degree of reproduction, resulting in banded and diagonally-dominant mass matrices. Dual basis functions derived from the inversion of the Gram matrix (IG) are the second type and already yield diagonal mass matrices. We will show that it is possible to apply the dual scheme as a transformation of the resulting system of equations based on NURBS for both – shape and test functions. Hence, it can be easily implemented into existing IGA routines and it is also promising to retain the accuracy known from similar formulations without mass lumping. Applying additional row-sum lumping to the mass matrices is either not necessary for IG or the caused loss of accuracy is lowered to a reasonable magnitude in the case of AD. Numerical examples show a significantly better approximation of the dynamic behavior for the dual lumping scheme compared to standard NURBS approaches making use of conventional row-sum lumping. In a nutshell, applying IG yields accurate numerical results but fully populated stiffness matrices occur, which are entirely unsuitable for explicit dynamic simulations, while combining AD and row-sum lumping leads to efficient dynamical computations, with respect to effort and accuracy. KW - Mass lumping scheme KW - Dual basis functions KW - Isogeometric analysis KW - Explicit dynamics KW - NURBS Y1 - 2024 U6 - https://doi.org/10.24352/UB.OVGU-2024-052 SN - 2199-9244 VL - Vol. 44 IS - 1 SP - 14 EP - 46 ER - TY - GEN A1 - Azizi, Nima A1 - Dornisch, Wolfgang T1 - A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition T2 - Computational Mechanics N2 - 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. Y1 - 2024 U6 - https://doi.org/10.1007/s00466-024-02444-w SN - 0178-7675 SP - 1 EP - 23 ER - TY - GEN A1 - Azizi, Nima A1 - Dornisch, Wolfgang T1 - Correction: A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition T2 - Computational Mechanics Y1 - 2024 U6 - https://doi.org/10.1007/s00466-024-02474-4 SN - 0178-7675 SP - S. 1 ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Azizi, Nima T1 - Vergleich zwischen isogeometrischen und spektralen Reissner-Mindlin Schalenelementen T2 - 15. Fachtagung Baustatik – Baupraxis, 4.-5. März 2024, Hamburg N2 - Der Beitrag vergleicht die Genauigkeit von Reissner-Mindlin-Schalenelementen zwischen der isogeometrischen Methode und der spektralen Elemente-Methode. Während die erste durch die hohe Kontinuität zwischen den Elementen eine sehr hohe Genauigkeit in Bezug auf Anzahl der Freiheitsgrade aufweist, besticht die letztere durch eine einfachere Formulierung und eine bessere Konditionierung auch für sehr hohe Ansatzordnungen. Y1 - 2024 UR - https://www.bb15.baustatik-baupraxis.de/Programm/program_presentation.php?id=65 ER - TY - GEN A1 - Stammen, Lisa A1 - Dornisch, Wolfgang T1 - On the use of mixed basis function degrees within a convective isogeometric element formulation T2 - Proceedings in applied mathematics and mechanics : PAMM N2 - Solving linear elasticity problems using standard finite element methods, different locking phenomena can occur. In order to counteract these effects, mixed methods or formulations using higher approximation orders can be employed, for instance. As a consequence, this leads to an increased computational effort. Hence, a selective elevation of orders in decisive directions within a purely displacement-based element formulation is proposed in this contribution. Within isogeometric analysis (IGA), the geometry is discretized using non-uniform rational B-splines (NURBS), which simultaneously represent the basis functions for the analysis. Due to the fact that the geometry can be preserved exactly during analysis, this can increase the accuracy of results. In this contribution, convective basis systems that are aligned with the local geometry are employed combined with selective order elevation. The required convective basis systems are interpolated from those determined in each control point. Y1 - 2023 U6 - https://doi.org/10.1002/pamm.202200159 SN - 1617-7061 N1 - 92nd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) VL - 23 IS - 1 ER - TY - GEN A1 - Stammen, Lisa A1 - Dornisch, Wolfgang T1 - Investigations on adapted interpolation orders for a mixed isogeometric plate formulation T2 - Proceedings in applied mathematics and mechanics : PAMM N2 - In order to overcome locking effects that especially occur for lower order finite element formulations, different methods can be employed. This can be conducted using mixed formulations or adapted approximation orders, for instance. Hence, in order to tackle shear locking that is caused by non-matching interpolation degrees in the shear strain equation, an irreducible and a mixed Reissner-Mindlin plate formulation with accordingly adapted conforming discretizations are derived within the scope of this contribution. In addition, non-uniform rational B-splines (NURBS) are employed therefore, in order to benefit from the properties and refinement strategies offered by isogeometric analysis (IGA) and to achieve more accurate results. The effect of various combinations of interpolation orders on the convergence behavior and the ability to alleviate locking is investigated for both the irreducible and the mixed isogeometric plate formulation and examined for a benchmark example. This is also supplemented by investigations on the stability of the considered variants, tested by the existence of the correct number of zero-energy modes. Y1 - 2023 U6 - https://doi.org/10.1002/pamm.202300170 SN - 1617-7061 N1 - 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), December 2023 VL - 23 IS - 4 ER -