TY - GEN A1 - Zou, Zhihui A1 - Scott, Michael A. A1 - Borden, Michael J. A1 - Thomas, Derek C. A1 - Dornisch, Wolfgang A1 - Brivadis, Ericka T1 - Isogeometric Bézier dual mortaring: Refineable higher-order spline dual bases and weakly continuous geometry T2 - Computer Methods in Applied Mechanics and Engineering N2 - In this paper we develop the isogeometric Bézier dual mortar method. It is based on Bézier extraction and projection and is applicable to any spline space which can be represented in Bézier form (i.e., NURBS, T-splines, LR-splines, etc.). The approach weakly enforces the continuity of the solution at patch interfaces and the error can be adaptively controlled by leveraging the refineability of the underlying slave dual spline basis without introducing any additional degrees of freedom. As a consequence, optimal higher-order convergence rates can be achieved without the need for an expensive shared master/slave segmentation step. We also develop weakly continuous geometry as a particular application of isogeometric Bézier dual mortaring. Weakly continuous geometry is a geometry description where the weak continuity constraints are built into properly modified Bézier extraction operators. As a result, multi-patch models can be processed in a solver directly without having to employ a mortaring solution strategy. We demonstrate the utility of the approach on several challenging benchmark problems. KW - Mortar methods KW - Isogeometric analysis KW - Bézier extraction KW - Bézier projection Y1 - 2018 U6 - https://doi.org/10.1016/j.cma.2018.01.023 SN - 0045-7825 VL - 333 SP - 497 EP - 534 ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Schrade, David A1 - Xu, Bai-Xiang A1 - Keip, Marc-André A1 - Müller, Ralf T1 - Coupled phase field simulations of ferroelectric and ferromagnetic layers in multiferroic heterostructures T2 - Archive of Applied Mechanics N2 - The combination of materials with either pronounced ferroelectric or ferromagnetic effect characterizes multiferroic heterostructures, whereby the different materials can be arranged in layers, columns or inclusions. The magnetization can be controlled by the application of electrical fields through a purely mechanical coupling at the interfaces between the different materials. Thus, a magneto-electric coupling effect is obtained. Within a continuum mechanics formulation, a phase field is used to describe the polarization and the magnetization in the ferroelectric and ferromagnetic layers, respectively. The coupling between polarization/magnetization and strains within the layers, in combination with the mechanical coupling at the sharp layer interfaces, yields the magneto-electric coupling within the heterostructure. The continuum formulations for both layers are discretized in order to make the differential equations amenable to a numerical solution with the finite element method. A state-of-the-art approach is used for the ferroelectric layer. The material behavior of the ferromagnetic layer is described by a continuum formulation from the literature, which is discretized using a newly proposed approach for the consistent interpolation of the magnetization vector. Four numerical examples are presented which show the applicability of the newly proposed approach for the ferromagnetic layer as well as the possibility to simulate magneto-electric coupling in multiferroic heterostructures. KW - Multiferroic heterostructure KW - Phase field method KW - Ferroelectric material KW - Ferromagnetic material KW - Finite element method KW - Rotation interpolation Y1 - 2019 U6 - https://doi.org/10.1007/s00419-018-1480-9 SN - 0939-1533 SN - 1432-0681 VL - 89 IS - 6 SP - 1031 EP - 1056 ER - TY - GEN A1 - Nadgir, Omkar A1 - Dornisch, Wolfgang A1 - Müller, Ralf A1 - Keip, Marc-André T1 - A phase-field model for transversely isotropic ferroelectrics T2 - Archive of Applied Mechanics N2 - We propose an electro-mechanically coupled phase-field model for ferroelectric materials that show cubic–tetragonal phase transition. The cubic phase is idealized by an isotropic formulation, and the tetragonal phase is idealized by a transversely isotropic formulation. We consider a classical phase-field model with Ginzburg–Landau-type evolution of the order parameter. The order parameter drives the transition of all involved moduli tensors such as elastic, dielectric and piezoelectric moduli, which in turn maintain their typical features and stability as a result of a selected phase-transition function. The model is described in coordinate-invariant form and implemented into a finite element framework with implicit time integration of the evolution equation. Representative numerical examples in two and three dimensions demonstrate the main features of the constitutive model and the numerical stability of the formulation. KW - Phase-field modeling KW - Ferroelectrics KW - Ginzburg–Landau equation KW - Transverse isotropy KW - Numerical simulations Y1 - 2019 U6 - https://doi.org/10.1007/s00419-019-01543-y SN - 0939-1533 SN - 1432-0681 VL - 89 IS - 6 SP - 1057 EP - 1068 ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Stöckler, Joachim A1 - Müller, Ralf T1 - Dual and approximate dual basis functions for B-splines and NURBS – Comparison and application for an efficient coupling of patches with the isogeometric mortar method T2 - Computer Methods in Applied Mechanics and Engineering N2 - This contribution defines and compares different methods for the computation of dual basis functions for B-splines and Non-Uniform Rational B-splines (NURBS). They are intended to be used as test functions for the isogeometric mortar method, but other fields of application are possible, too. Three different concepts are presented and compared. The first concept is the explicit formula for the computation of dual basis functions for NURBS proposed in the work of Carl de Boor. These dual basis functions entail minimal support, i.e., the support of the dual basis functions is equal to the support of the corresponding B-spline basis functions. In the second concept dual basis functions are derived from the inversion of the Gram matrix. These dual basis functions have global support along the interface. The third concept is the use of approximate dual basis functions, which were initially proposed for the use in harmonic analysis. The support of these functions is local but larger than the support of the associated B-spline basis functions. We propose an extension of the approximate dual basis functions for NURBS basis functions. After providing the general formulas, we elaborate explicit expressions for several degrees of spline basis functions. All three approaches are applied in the frame of the mortar method for the coupling of non-conforming NURBS patches. A method which allows complex discretizations with multiple intersecting interfaces is presented. Numerical examples show that the explicitly defined dual basis functions with minimal support severely deteriorate the global stress convergence behavior of the mechanical analysis. This fact is in accordance with mathematical findings in literature, which state that the optimal reproduction degree of arbitrary functions is not possible without extending the support of the dual basis functions. The dual basis functions computed from the inverse of the Gram matrix yield accurate numerical results but the global support yields significantly higher computational costs in comparison to computations of conforming meshes. Only the approximate dual basis functions yield accurate and efficient computations, where neither accuracy nor efficiency is significantly deteriorated in comparison to computations of conforming meshes. All basic cases of T-intersections and star-intersections are studied. Furthermore, an example which combines all basic cases in a complex discretization is given. The applicability of the presented method for the nonlinear case and for shell formulations is shown with the help of one numerical example. KW - Isogeometric analysis KW - Dual basis functions for NURBS KW - Optimal convergence KW - Mortar method KW - Coupling of non-conforming meshes KW - Approximate dual basis functions for NURBS Y1 - 2017 U6 - https://doi.org/10.1016/j.cma.2016.07.038 SN - 0045-7825 VL - 316 SP - 449 EP - 496 ER - TY - GEN A1 - Gravenkamp, Hauke A1 - Natarajan, Sundarajan A1 - Dornisch, Wolfgang T1 - On the use of NURBS-based discretizations in the scaled boundary finite element method for wave propagation problems T2 - Computer Methods in Applied Mechanics and Engineering N2 - We discuss the application of non-uniform rational B-splines (NURBS) in the scaled boundary finite element method (SBFEM) for the solution of wave propagation problems at rather high frequencies. We focus on the propagation of guided waves along prismatic structures of constant cross-section. Comparisons are made between NURBS-based discretizations and high-order spectral elements in terms of the achievable convergence rates. We find that for the same order of shape functions, NURBS can lead to significantly smaller errors compared with Lagrange polynomials. The difference becomes particularly important at very high frequencies, where spectral elements are prone to instabilities. Furthermore, we analyze the behavior of NURBS for the discretization of curved boundaries, where the benefit of exact geometry representation becomes crucial even in the low-frequency range. KW - Isogeometric analysis KW - caled boundary finite element method KW - Wave propagation KW - Spectral elements KW - NURBS Y1 - 2017 U6 - https://doi.org/10.1016/j.cma.2016.11.030 SN - 0045-7825 VL - 315 SP - 867 EP - 880 ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Stöckler, Joachim A1 - Müller, Ralf T1 - Recent advances in isogeometric dual mortar patch coupling T2 - Proceedings of the 7th GACM Colloquium on Computational Mechanics for Young Scientists from Academia and Industry, October 11-13, 2017 in Stuttgart, Germany N2 - Isogeometric analysis fosters the integration of design and analysis by using the geometry description of the CAD system also for the numerical analysis. Hereby, the use of NURBS surfaces is common but entails the need for a coupling of non-conforming patches. The use of mortar methods allows a coupling which requires neither additional variables nor empirical parameters. In this contribution dual basis functions are used in order to obtain an accurate and efficient mortar method. Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bsz:93-opus-ds-93516 SP - 467 EP - 470 PB - Institute for Structural Mechanics University of Stuttgart CY - Stuttgart ER - TY - GEN A1 - Kikis, Georgia A1 - Dornisch, Wolfgang A1 - Klinkel, Sven T1 - A method for the elimination of shear locking effects in an isogeometric Reissner-Mindlin shell formulation T2 - Proceedings of the 7th GACM Colloquium on Computational Mechanics for Young Scientists from Academia and Industry, October 11-13, 2017 in Stuttgart, Germany N2 - Shell elements for slender structures based on a Reissner-Mindlin approach struggle in pure bending problems. The stiffness of such structures is overestimated due to the transversal shear locking effect. Here, an isogeometric Reissner-Mindlin shell element is presented, which uses adjusted control meshes for the displacements and rotations in order to create a conforming interpolation of the pure bending compatibility requirement. The method is tested for standard numerical examples. Y1 - 2017 U6 - https://doi.org/10.18419/opus-9334 SP - 500 EP - 503 PB - Institute for Structural Mechanics University of Stuttgart CY - Stuttgart ER - TY - GEN A1 - Kikis, Georgia A1 - Dornisch, Wolfgang A1 - Klinkel, Sven T1 - Isogeometric Reissner-Mindlin shell analysis - adjusted approximation spaces for the reduction of shear locking effects T2 - VI International Conference on Isogeometric Analysis N2 - In this contribution, an isogeometric Reissner-Mindlin shell element is presented, which uses adjusted approximation spaces for the displacements and the rotations in order to avoid transversal shear locking effects. These locking effects arise especially in pure bending problems, with decreasing thickness of the shell. Their origin lies in the not conforming interpolations of the displacements and rotations in the formulation of the compatibility requirements. One possibility to overcome this difficulty would be to increase the polynomial degree of the used NURBS shape functions, as it is proposed in [1]. However, the locking effects are not completely eliminated and the computational time for the formation of the stiffness matrix increases significantly with rising polynomial degrees. For lower polynomial degrees, there exist only a few effective concepts for the prevention of locking. Beir˜ao da Veiga and his group proposed one of these methods for the elimination of transversal shear locking effects for plate problems [2]. They suggested the implementation of different control meshes, with adjusted polynomial degrees for the interpolation of the displacements and the rotations. In this way transversal shear locking effects due to the coupling of shear strains and curvature are avoided. The isogeometric concept still holds because the reference geometry for the displacements and the rotations is the same and only the refinement is different. This method is now extended to an isogeometric Reissner-Mindlin shell formulation. The used shell element is derived from continuum theory. Nodal basis systems are computed with a globalL2 fit and thus the interpolated director vectors of the reference configuration coincide best possible with the normal vectors. The interpolation of the current director vector is performed using a full SO(3) update, so the nodal rotations are interpolated. This deviates from standard rotation interpolation procedures, where the current director vector is interpolated. The more accurate interpolation used in this contribution leads to more accurate results, see [3]. The accuracy and efficiency of the shell element are examined for some standard benchmark examples. Y1 - 2017 UR - http://congress.cimne.com/iga2017/admin/files/fileabstract/a74.pdf PB - CIMNE CY - Barcelona ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Schrade, David A1 - Xu, Bai-Xiang A1 - Müller, Ralf T1 - Coupling of phase field models for ferroelectric and ferromagnetic layers in multiferroic heterostructures T2 - VII International Conference on Coupled Problems in Science and Engineering (Coupled Problems 2017) N2 - Heterostructures of ferroelectric and ferromagnetic layers are commonly used to obtain electromagnetic effects. The elastic coupling between the layers is widely acknowledged as the main mechanism responsible for the electro-magneto interaction. Within this contribution we study the coupling of ferroelectric and ferromagnetic layers with well-defined interfaces. The intention is to simulate the switching of the magnetization with the help of electric fields, which has been studied experimentally in [1]. Each layer is simulated by using mechanically coupled phase field modeling, whereby the approaches presented in [2] and [3] will be used. The strains in each layer depend on the direction of the polarization/magnetization. A mismatch of these strains will be compensated by local deformations at the interface as the coupling results from the coherent deformation at the interface. This leads to the possibility to alter the magnetization direction by changing the electric polarization and vice verse. Numerical simulations will illustrate the evolution of the ferroic microstructures with a focus on the strain coupling and the resulting interactions between layers. Y1 - 2017 UR - http://congress.cimne.com/coupled2017/admin/files/fileabstract/a365.pdf PB - CIMNE CY - Barcelona ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Stöckler, Joachim T1 - An isogeometric mortar method for the coupling of multiple NURBS domains with optimal convergence rates T2 - Numerische Mathematik N2 - We investigate the mortar finite element method for second order elliptic boundary value problems on domains which are decomposed into patchesk with tensor-product NURBS parameterizations. We follow the methodology of IsoGeometric Analysis (IGA) and choose discrete spaces Xh,k on each patch k as tensor-product NURBS spaces of the same or higher degree as given by the parameterization. Our work is an extension of Brivadis et al. (Comput Methods Appl Mech Eng 284:292–319, 2015) 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 Xh,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. KW - Isogeometric analysis KW - Mortar Method KW - Approximate duals KW - domain coupling Y1 - 2021 U6 - https://doi.org/10.1007/s00211-021-01246-z SN - 0945-3245 SN - 0029-599X VL - 149 IS - 4 SP - 871 EP - 931 ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Stöckler, Joachim T1 - An isogeometric mortar method with optimal convergence and reduced support T2 - 24th International Conference on Computer Methods in Mechanics (CMM) and the 42nd Solid Mechanics Conference (SolMech) N2 - The main feature of isogeometric analysis is the usage of a coherent geometry description for design and analysis. In most cases, Non-Uniform Rational B-splines (NURBS) are used within the frame of the finite element method, which combines the most common geometrical modeling concept with the most common analysis method in structural mechanics. NURBS are a versatile tool for geometric modeling, and in order to define complex geometric structures, a multitude of tensor-product NURBS patches is required. Typical software in Computer-Aided design can manage smoothness requirements across common interfaces of neighboring patches. For the finite element method, a different type of coupling across interfaces must be achieved. Due to the nature of tensor product NURBS, the simple concept of mutual refinement and subsequent coupling by shared degrees of freedom is prohibitively costly and, in some cases, even not possible. A multitude of coupling methods has been proposed over the last years. The most common concepts are known as mortar methods. In particular, the dual mortar method has been shown to yield very efficient computations. A recent paper by the authors has provided an isogeometric mortar method with mathematically proven optimal convergence of the stress errors over the entire domain. We use dual basis functions, which have support only on one interface and avoid interrelations between different interfaces. Models with a large number of intersecting interfaces can be handled. However, the basis functions have full support on the interfaces. In our current contribution, we propose the use of approximate dual basis functions with the advantage of having local support on the interfaces. These functions fulfill the duality only in an approximate way, but still guarantee the optimal degree for the convergence of the mortar method. Since the duality is not fulfilled, an additional lumping of the mortar matrix is introduced. The error of this lumping can be analyzed mathematically and is not significant in comparison to the global approximation error of the finite element method. The use of the approximate dual basis functions restores the local support of basis functions along the interface while the convergence properties remain intact. Numerical examples show the convergence behavior for simple and complex models. Y1 - 2022 UR - http://cmm-solmech.ippt.pan.pl/S01.html#ID_119 ER - TY - THES A1 - Dornisch, Wolfgang T1 - Interpolation of Rotations and Coupling of Patches in Isogeometric Reissner– Mindlin Shell Analysis N2 - This work is concerned with the development of an efficient and robust isogeometric Reissner–Mindlin shell formulation. The basic assumption of shell theories is a dimensional reduction of the three-dimensional continuum to a two-dimensional surface embedded in the three-dimensional space. Consequently, the geometry is described by a reference surface in combination with a director vector field, which defines the expansion in the thickness direction. The main objective of isogeometric analysis is to use the same model description for design and analysis. Thin-walled structures are usually defined by a reference surface and an associated thickness in industrial design software. Thus, the usage of isogeometric shell elements can avoid costly conversions to volumetric geometry descriptions.The usage of NURBS surfaces (Non-Uniform Rational B-splines) possibly yields high continuity between elements. This requires a rethinking of all concepts used in conventional shell elements, which base on linear Lagrange basis functions. The shell formulation presented in this work is derived from the continuum theory and uses an orthogonal rotation described by Rodrigues' tensor to compute the current director vector. Large deformations and finite rotations can be described accurately. The discretization requires nodal director vectors which interpolate the normal vector as exact as possible. A new method for the definition of nodal basis systems and nodal director vectors is derived. Basing on this, a criterion for the automatic assignment of the correct number of rotational degrees of freedom for each node is proposed. This allows stable computations of geometries with kinks while requiring neither the usage of drilling rotation stabilization nor manual user interaction. The main part of this work is the derivation of various concepts for the interpolation of the current director vector, which is a function of the rotational state. The respective concepts differ in the quantity which is actually interpolated and in the chosen update formulation for the rotations. The influence of each concept on the global deformation convergence behavior is assessed with the help of numerical examples. The results suggest that proper convergence behavior for all orders of NURBS basis functions can only be attained if interpolated director vectors are rotated. Concepts of this type are more accurate and expensive than concepts which rotate nodal director vectors. But the higher computational effort pays off for geometries with arbitrary curvature and for basis functions of higher order. Geometries with kinks require a multiplicative rotational update formulation for concepts that rotate interpolated director vectors.Three different integration rules are considered within the numerical examples. Besides full and reduced Gauss integration also a new non-uniform Gauss integration concept following Adam et al. (2015) is assessed. A special focus is put on the interaction between the chosen rotational concept and the integration scheme. The reduction of the number of integration points from full to reduced integration slightly alleviates locking effects. The further reduction entailed by non-uniform integration significantly reduces locking effects in some examples. But this only yields higher accuracy if a concept which rotates interpolated director vectors is chosen. The reduction of locking deteriorates the accuracy of deformation results in other cases. The efficiency of the presented shell formulation is compared to standard shell formulations in terms of computational costs to attain a pre-defined error level. The most effective combination of integration scheme and rotational concept is shown to be competitive to standard shell formulations.A further main concern of this work is the derivation of a mortar-type method for the coupling of non-conforming NURBS surface patches. Methods to handle non-conforming patches without mutual refinement are essential for an efficient application of NURBS-based isogeometric analysis. The proposed method bases on a substitution relation, which is derived from the weak fulfillment of the equality of mutual displacements along the interface. A static condensation can be performed with the help of this substitution relation in order to attain a coupled global system of equations. The variational formulation is not altered and the global stiffness matrix remains positive definite. Numerical examples show the applicability of the method. A comparison to reference results and to computations with the Lagrange multiplier method is given. The applicability of the coupling method for the presented Reissner-Mindlin shell formulation is shown with the help of two nonlinear examples. N2 - Die vorliegende Arbeit befasst sich mit der Entwicklung einer effizienten und robusten isogeometrischen Reissner-Mindlin-Schalenformulierung. Die grundlegende Annahme bei Schalentheorien ist eine Dimensionsreduzierung des dreidimensionalen Kontinuums auf eine zweidimensionale Fläche im dreidimensionalen Raum. Somit wird die Geometrie durch eine Referenzfläche in Verbindung mit einem Direktor-Vektor-Feld beschrieben. Die Ausdehnung in Dickenrichtung wird durch die Direktor-Vektoren definiert. Das Hauptziel der isogeometrischen Analyse ist es, die gleiche Modell-Beschreibung für Entwurf und Berechnung zu verwenden. Dünnwandige Strukturen werden in Entwurfsprogrammen durch eine Referenzfläche mitsamt zugehöriger Dicke beschrieben. Somit kann durch die Verwendung isogeometrischer Schalenelemente die aufwändige Umrechnung in Volumen-Geometriebeschreibungen vermieden werden.Die Verwendung von NURBS-Flächen (Non-Uniform Rational B-splines) kann zu hoher Kontinuität an den Elementgrenzen führen. Dies erfordert ein Überdenken aller Konzepte, die in konventionellen Schalenelementen, welche auf linearen Lagrange-Basisfunktionen basieren, verwendet werden. Die in dieser Arbeit vorgestellte Schalenformulierung ist aus der Kontinuumstheorie hergeleitet und verwendet eine orthogonale Drehung, welche durch den Drehtensor nach Rodrigues beschrieben wird, zur Berechnung des Direktor-Vektors in der Momentankonfiguration. Große Verformungen und endliche Verdrehungen können präzise beschrieben werden.Für die Diskretisierung werden Direktor-Vektoren in den Knoten benötigt. Diese sollen die Normalenvektoren so genau wie möglich interpolieren. Eine neue Methode zur Bestimmung von Knoten-Basissystemen und Knoten-Direktor-Vektoren wird hergeleitet. Darauf aufbauend wird ein Kriterium vorgeschlagen, welches eine automatische Bestimmung der passenden Anzahl von Rotationsfreiheitsgraden für jeden Knoten ermöglicht. Dadurch sind stabile Berechnungen von Geometrien mit Knicken möglich, ohne dass die Verwendung von Methoden zur Drill-Rotations-Stabilisierung oder ein manueller Eingriff des Benutzers notwendig sind.Die Herleitung verschiedener Konzepte für die Interpolation des Direktor-Vektors in der Momentankonfiguration stellt den Hauptteil dieser Arbeit dar. Die Momentankonfiguration des Direktor-Vektors ist eine Funktion des Verdrehzustands. Die jeweiligen Konzepte unterscheiden sich durch die Größe, welche interpoliert wird, sowie durch die gewählte Update-Formulierung für die Verdrehungen. Der Einfluss jedes Konzepts auf das globale Deformations-Konvergenzverhalten wird mit Hilfe von numerischen Beispielen untersucht. Die Ergebnisse legen nahe, dass ordnungsgemäßes Konvergenzverhalten für alle Grade von NURBS-Basisfunktionen nur erreicht werden kann, falls interpolierte Direktor-Vektoren verdreht werden. Konzepte dieser Art sind genauer und aufwendiger als Konzepte, die Knoten-Direktor-Vektoren verdrehen. Aber der höhere Berechnungsaufwand zahlt sich bei Geometrien mit beliebiger Krümmung und bei Basisfunktionen höheren Grades aus. Geometrien mit Knicken erfordern eine multiplikative Update-Formulierung für die Verdrehungen, falls ein Konzept verwendet wird, das interpolierte Direktor-Vektoren verdreht.Drei verschiedene Integrationsregeln werden in den numerischen Beispielen berücksichtigt. Neben vollständiger und reduzierter Integration nach Gauß wird auch ein neues nicht-gleichförmiges Integrationsschema in Anlehnung an Adam et al. (2015) untersucht. Ein besonderer Schwerpunkt liegt auf der gegenseitigen Beeinflussung von gewähltem Rotationskonzept und Integrationsschema. Die Reduzierung der Anzahl der Integrationspunkte von vollständiger zu reduzierter Integration führt zu einer leichten Verminderung von Versteifungseffekten. Die weitere Verringerung durch die Anwendung des nicht-gleichförmigen Integrationsschemas führt bei manchen Beispielen zu einer erheblichen Reduzierung von Versteifungseffekten. Dies führt jedoch nur zu höherer Genauigkeit falls ein Konzept verwendet wird, das interpolierte Direktor-Vektoren verdreht. In anderen Fällen führt die Verringerung von Versteifungseffekten zu einer Abnahme der Genauigkeit der Verschiebungsergebnisse. Die Effizienz der vorgestellten Schalenformulierung wird anhand der Berechnungskosten, die zum Erreichen eines vordefinierten Fehlerniveaus notwendig sind, mit konventionellen Schalenformulierungen verglichen. Hierbei zeigt sich, dass die effektivste Kombination von Integrationsschema und Rotationskonzept konkurrenzfähig zu konventionellen Schalenformulierungen ist.Ein weiteres Hauptanliegen dieser Arbeit ist die Herleitung einer Mortar-Methode zur Kopplung nicht-konformer NURBS-Flächenpatches. Methoden zur Berechnung nicht-konformer Patches ohne gegenseitige Netzverfeinerung sind unerlässlich für eine wirtschaftliche Anwendung von NURBS-basierten isogeometrischen Methoden. Das vorgestellte Verfahren basiert auf einer Substitutions-Beziehung, welche aus der schwachen Erfüllung der Gleichheit der gegenseitigen Verschiebungen entlang der Verbindungslinie hergeleitet wird. Mit Hilfe dieser Beziehung kann durch eine statische Kondensation das globale Gleichungssystem gekoppelt werden. Die Variationsformulierung wird nicht verändert und die globale Steifigkeitsmatrix bleibt positiv definit. Anhand numerischer Beispiele wird die Anwendbarkeit der Methode aufgezeigt. Die Ergebnisse werden mit Referenzergebnissen und mit Berechnungen mit der Lagrange-Multiplikator-Methode verglichen. Die Anwendbarkeit der Kopplungs-Methode für die vorgestellte Reissner-Mindlin-Schalenformulierung wird mit Hilfe zweier nichtlinearer Beispiele gezeigt. KW - Isogeometric Analysis KW - Reissner-Mindlin Shell Analysis KW - NURBS KW - Interpolation of Rotations KW - Integration schemes KW - Domain Decomposition KW - Mortar Method KW - Weak Substitution Method Y1 - 2015 UR - http://d-nb.info/1126124044/34 UR - http://publications.rwth-aachen.de/record/466744/files/466744.pdf SN - 978-3-946090-02-1 PB - Schriftenreihe des Lehrstuhls für Baustatik und Baudynamik der RWTH Aachen CY - Aachen ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Yan, Sikang ED - Bischoff, Manfred ED - Scheven, Malte von ED - Oesterle, Bastian T1 - Effiziente Integrationsmethoden für isogeometrische Schalenelemente T2 - Berichte der Fachtagung Baustatik – Baupraxis 14 N2 - Die isogeometrische Methode definiert sich durch den Einsatz einer einheitlichen Geometriebeschreibung für Entwurf und Berechnung. Insbesondere bei der Berechnung dünnwandiger Flächentragwerke ist durch die Verwendung der exakten Geometrie ein großer Gewinn an Genauigkeit und Zuverlässigkeit möglich. Um neben hoher Genauigkeit auch hohe Effizienz zu erreichen, wird in diesem Beitrag die Verwendung effizienter numerischer Integrationsmethoden für die Berechnung der Steifigkeitsmatrix untersucht. KW - Isogeometric analysis KW - NURBS basis functions KW - Efficient Integration Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bsz:93-opus-ds-107793 UR - http://elib.uni-stuttgart.de/handle/11682/10779 SN - 978-3-00-064639-3 SP - 415 EP - 422 PB - Institut für Baustatik und Baudynamik, Universität Stuttgart CY - Stuttgart ER - TY - GEN A1 - Schmidt, Simon David A1 - Ammar, Kais A1 - Dornisch, Wolfgang A1 - Forest, Samuel A1 - Müller, Ralf T1 - Phase field model for the martensitic transformation: comparison of the Voigt/Taylor and Khachaturyan approach T2 - Continuum Mechanics and Thermodynamics Y1 - 2021 U6 - https://doi.org/10.1007/s00161-021-01007-1 SN - 0935-1175 SN - 1432-0959 VL - 33 IS - 5 SP - 2075 EP - 2094 ER - TY - GEN A1 - Stammen, Lisa A1 - Dornisch, Wolfgang ED - Korobenko, Artem ED - Evans, John ED - Hsu, Ming-Chen T1 - A convective isogeometric element formulation with mixed basis function degrees T2 - Isogeometric Analysis 2022 (IGA 2022) - Book of Abstracts N2 - Employing isogeometric analysis (IGA), the geometry of structures is discretized by non-uniform rational B-splines (NURBS), which simultaneously provide the basis functions for the corresponding analysis as well. Due to the high continuity within patches, a unique local convective basis system, which corresponds to the local geometry directions of the mesh, can be defined in every point. Locking Phenomena, which strongly affect purely displacement-based low order elements, can be counteracted by employing particular methods or higher polynomial degrees; both options increase the resulting computational effort significantly. Using order elevation only in specific directions has the potential to optimize the ratio between locking counteraction and computational costs. In this contribution, the use of directed deformations based on convective basis systems in each control point is proposed for a displacement-based isogeometric formulation with specifically adapted orders. Therefore, distinct meshes for the interpolation of the displacements in each direction are generated based on the initial geometry. Subsequently, the order of every mesh is elevated in only one direction. Consequently, different possibilities for the combination of order elevations have to be examined. This procedure is conducted for a two-dimensional linear elasticity problem. The benefit of a convective formulation with direction dependent degrees is shown by a comparison to a standard isogeometric formulation. Furthermore, the impact of the directions selected for order elevation on the accuracy of the results is investigated. Y1 - 2022 UR - https://drive.google.com/file/d/1CYA-jTeVREnLcY2XIY1HuYiYutElGxXE/view SP - 103 ER - TY - GEN A1 - Zou, Zhihui A1 - Scott, Michael A. A1 - Miao, Di A1 - Bischoff, Manfred A1 - Oesterle, Bastian A1 - Dornisch, Wolfgang T1 - An isogeometric Reissner–Mindlin shell element based on Bézier dual basis functions: Overcoming locking and improved coarse mesh accuracy T2 - Computer Methods in Applied Mechanics and Engineering N2 - We develop a mixed geometrically nonlinear isogeometric Reissner–Mindlin shell element for the analysis of thin-walled structures that leverages Bézier dual basis functions to address both shear and membrane locking and to improve the quality of computed stresses. The accuracy of computed solutions over coarse meshes, that have highly non-interpolatory control meshes, is achieved through the application of a continuous rotational approach. The starting point of the formulation is the modified Hellinger–Reissner variational principle with independent displacement, membrane, and shear strains as the unknown fields. To overcome locking, the strain variables are interpolated with lower-order spline bases while the variations of the strain variables are interpolated with the corresponding Bézier dual bases. Leveraging the orthogonality property of the Bézier dual basis, the strain variables are condensed out of the system with only a slight increase in the bandwidth of the resulting linear system. The condensed approach preserves the accuracy of the non-condensed mixed approach but with fewer degrees of freedom. From a practical point of view, since the Bézier dual basis is completely specified through Bézier extraction, any spline space that admits Bézier extraction can utilize the proposed approach directly. KW - Isogeometric analysis KW - Reissner–Mindlin shells KW - Dual basis functions KW - Locking Y1 - 2020 U6 - https://doi.org/10.1016/j.cma.2020.113283 SN - 0045-7825 VL - 370 ER - TY - GEN A1 - Chasapi, Margarita A1 - Dornisch, Wolfgang A1 - Klinkel, Sven T1 - Patch coupling in isogeometric analysis of solids inboundary representation using a mortar approach T2 - International Journal for Numerical Methods in Engineering N2 - This contribution is concerned with a coupling approach for nonconforming NURBS patches in the framework of an isogeometric formulation for solids inboundary representation. The boundary representation modeling technique inCAD is the starting point of this approach. We parameterize the solid according to the scaled boundary finite element method and employ NURBS basis functions for the approximation of the solution. Therefore, solid surfaces consist of several sections, which can be regarded as patches and discretized independently. The main objective of this study is to derive an approach for the connection of independent sections in order to allow for local refinement and thus an accurate and efficient discretization of the computational domain. Non-conforming sections are coupled with a mortar approach within a master-slave framework. The coupling of adjacent sections ensures the equality of mutual deformations along the interface in a weak sense and is enforced by constraining the NURBS basis functions on the interface. We apply this approach to non-linear problems in two dimensions and compare the results with conforming discretizations. KW - Isogeometric analysis KW - Scaled Boundary FEM KW - Mortar Method Y1 - 2020 U6 - https://doi.org/10.1002/nme.6354 SN - 1097-0207 VL - 121 IS - 14 SP - 3206 EP - 3226 ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Klinkel, Sven A1 - Simeon, Bernd T1 - Isogeometric Reissner–Mindlin shell analysis with exactly calculated director vectors T2 - Computer Methods in Applied Mechanics and Engineering N2 - 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. KW - Isogeometric analysis KW - Reissner–Mindlin shell KW - NURBS KW - Interpolation of the director Y1 - 2013 U6 - https://doi.org/10.1016/j.cma.2012.09.010 SN - 0045-7825 VL - 253 SP - 491 EP - 504 ER - TY - GEN A1 - Klinkel, Sven A1 - Chen, Lin A1 - Dornisch, Wolfgang T1 - A NURBS based hybrid collocation-Galerkin method for the analysis of boundary represented solids T2 - Computer Methods in Applied Mechanics and Engineering N2 - The paper is concerned with a new numerical method, NURBS based hybrid collocation–Galerkin method (NURBS-HCGM), to solve the in-plane motion problem of elastic solids. It combines the merits of the so-called scaled boundary finite-element method (SB-FEM) and the isogeometric collocation method. For the analysis, the boundary scaling technique of SB-FEM is adopted. It leads to a formulation, where only the boundary of a structure is discretized. Here, the NURBS basis functions are employed for the description of the geometry of the boundary as well as for the approximation of the displacements at the boundary. This is in accordance with the boundary representation modeling technique, which is commonly employed in computer aided design software. The inner domain is described by a radial scaling parameter. Applying the weak form only in circumferential direction the governing partial differential equations of elasticity are transformed to an ordinary differential equation (ODE) of Euler type, where the unknown displacements are a function of the radial scaling parameter. In the present work a NURBS based collocation scheme is introduced to solve this equation. NURBS basis functions are suggested for the approximation of the displacements in scaling direction. The higher continuity provided by NURBS allows to use collocation to solve the ODE directly instead of using the weak form in scaling direction. The proposed approach is validated by comparison with the eigenvalue solution of the ODE. It is remarked that the eigenvalue solution is restricted to linear problems, whereas the proposed method could be extended to nonlinear problems. In general, the presented formulation will allow to model patches bounded by an arbitrary number of contour boundaries. The accuracy of the proposed approach is analyzed and estimated with respect to analytical solutions. The computational cost is investigated with the help of numerical examples and is compared to isogeometric Galerkin approach. KW - Collocation method KW - NURBS basis functions KW - Scaled boundary finite element method KW - Hybrid collocation–Galerkin method Y1 - 2015 U6 - https://doi.org/10.1016/j.cma.2014.10.029 SN - 0045-7825 VL - 284 SP - 689 EP - 711 ER - TY - GEN A1 - Kikis, Georgia A1 - Dornisch, Wolfgang A1 - Klinkel, Sven T1 - Adjusted approximation spaces for the treatment of transverse shear locking in isogeometric Reissner–Mindlin shell analysis T2 - Computer Methods in Applied Mechanics and Engineering N2 - Transverse shear locking is an issue that occurs in Reissner–Mindlin plate and shell elements. It leads to an artificial stiffening of the system and to oscillations in the stress resultants for thin structures. The thinner the structure is, the more pronounced are the effects. Since transverse shear locking is caused by a mismatch in the approximation spaces of the displacements and the rotations, a field-consistent approach is proposed for an isogeometric degenerated Reissner–Mindlin shell formulation. The efficiency and accuracy of the method is investigated for benchmark plate and shell problems. A comparison to element formulations with locking alleviation methods from the literature is provided. KW - Isogeometric analysis KW - Reissner–Mindlin shells KW - Transverse shear locking KW - Adjusted approximation spaces Y1 - 2019 U6 - https://doi.org/10.1016/j.cma.2019.05.037 SN - 0045-7825 VL - 354 SP - 850 EP - 870 ER - TY - GEN A1 - Sobota, Paul M. A1 - Dornisch, Wolfgang A1 - Müller, Ralf A1 - Klinkel, Sven T1 - Implicit dynamic analysis using an iso- geometric Reissner–Mindlin shell formulation T2 - International Journal for Numerical Methods in Engineering N2 - In isogeometric analysis, identical basis functions are used for geometrical representation and analysis. In this work, non‐uniform rational basis splines basis functions are applied in an isoparametric approach. An isogeometric Reissner–Mindlin shell formulation for implicit dynamic calculations using the Galerkin method is presented. A consistent as well as a lumped matrix formulation is implemented. The suitability of the developed shell formulation for natural frequency analysis is demonstrated by a numerical example. In a second set of examples, transient problems of plane and curved geometries undergoing large deformations in combination with nonlinear material behavior are investigated. Via a zero‐thickness stress algorithm for arbitrary material models, a J2‐plasticity constitutive law is implemented. In the numerical examples, the effectiveness, robustness, and superior accuracy of a continuous interpolation method of the shell director vector is compared with experimental results and alternative numerical approaches. KW - isogeometric analysis KW - Reissner–Mindlin shell KW - NURBS KW - nonlinear material behavior KW - Structural vibrations KW - implicit dynamics KW - continuous director vector interpolation Y1 - 2017 U6 - https://doi.org/10.1002/nme.5429 SN - 1097-0207 VL - 110 IS - 9 SP - 803 EP - 825 ER - TY - GEN A1 - Aziz Ali, Sk A1 - Yan, Sikang A1 - Dornisch, Wolfgang A1 - Stricker, Didier T1 - Foldmatch: Accurate and High Fidelity Garment Fitting Onto 3D Scans T2 - 2020 IEEE International Conference on Image Processing (ICIP) N2 - In this paper, we propose a new template fitting method that can capture fine details of garments in target 3D scans of dressed human bodies. Matching the high fidelity details of such loose/tight-fit garments is a challenging task as they express intricate folds, creases, wrinkle patterns, and other high fidelity surface details. Our proposed method of non-rigid shape fitting – FoldMatch – uses physics-based particle dynamics to explicitly model the deformation of loose-fit garments and wrinkle vector fields for capturing clothing details. The 3D scan point cloud behaves as a collection of astrophysical particles, which attracts the points in template mesh and defines the template motion model. We use this point-based motion model to derive regularized deformation gradients for the template mesh. We show the parameterization of the wrinkle vector fields helps in the accurate shape fitting. Our method shows better performance than the stateof-the-art methods. We define several deformation and shape matching quality measurement metrics to evaluate FoldMatch on synthetic and real data sets. KW - Shape Matching KW - Wrinkle Vector Field KW - Cloth Simulation KW - Deformation Gradient KW - N-body Problem KW - Three-dimensional displays Y1 - 2020 UR - https://ieeexplore.ieee.org/document/9190730 SN - 978-1-7281-6395-6 U6 - https://doi.org/10.1109/ICIP40778.2020.9190730 SN - 2381-8549 CY - Abu Dhabi, United Arab Emirates ER - TY - GEN A1 - Chen, Lin A1 - Klinkel, Sven A1 - Dornisch, Wolfgang T1 - Hybrid collocation-Galerkin approach for the analysis of surface represented 3D-solids employing SB-FEM T2 - Computer Methods in Applied Mechanics and Engineering N2 - This paper presents a numerical method to solve the three-dimensional elasticity problem of surface represented solids. A surface oriented formulation is derived, in which the three-dimensional solid is described by its boundary surfaces and a radial scaling center. Scaling the boundary surfaces with respect to the scaling center yields a parameterization of the complete solid. The definition of the boundary surfaces is sufficient for the description of the solid. Thus, the formulation conforms ideally to the boundary representation modeling technique used in CAD. In the present approach, no tensor-product structure of three-dimensional objects is exploited to parameterize the physical domain. The weak form of the equation of motion and the weak form of the Neumann boundary conditions are enforced only at the boundary surfaces. It leads to a transformation of the governing partial differential equations of elasticity to an ordinary differential equation (ODE) of Euler type. Solving the ODE leads to the displacement in the radial scaling direction. In the present approach, the isogeometric Galerkin finite element method is employed to describe the geometry of the boundary surfaces and also to approximate the displacement response of the boundary surfaces. It exploits two-dimensional NURBS objects to parameterize the boundary surfaces. The final Euler type ODE is solved by the NURBS based collocation method. The displacement response in the radial scaling direction is approximated by the NURBS basis functions. Hence, the method can be extended to nonlinear problems. The accuracy of the method is validated against analytical solutions. In general, the proposed formulation is able to model solids bounded by an arbitrary number of surfaces. KW - Surface represented solid KW - NURBS basis functions KW - Scaled boundary finite element method KW - Hybrid collocation-Galerkin method Y1 - 2015 U6 - https://doi.org/10.1016/j.cma.2015.07.004 SN - 0045-7825 VL - 295 SP - 268 EP - 289 ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Müller, Ralf A1 - Klinkel, Sven T1 - An efficient and robust rotational formulation for isogeometric Reissner–Mindlin shell elements T2 - Computer Methods in Applied Mechanics and Engineering N2 - This work is concerned with the development of an efficient and robust isogeometric Reissner–Mindlin shell formulation for the mechanical simulation of thin-walled structures. Such structures are usually defined by non-uniform rational B-splines (NURBS) surfaces in industrial design software. The usage of isogeometric shell elements can avoid costly conversions from NURBS surfaces to other surface or volume geometry descriptions. The shell formulation presented in this contribution uses a continuous orthogonal rotation described by Rodrigues’ tensor in every integration point to compute the current director vector. The rotational state is updated in a multiplicative manner. Large deformations and finite rotations can be described accurately. The proposed formulation is robust in terms of stable convergence behavior in the nonlinear equilibrium iteration for large load steps and geometries with large and arbitrary curvature, and in terms of insensitivity to shell intersections with kinks under small angles. Three different integration schemes and their influence on accuracy and computational costs are assessed. The efficiency and robustness of the proposed isogeometric shell formulation is shown with the help of several examples. Accuracy and efficiency is compared to an isogeometric shell formulation with the more common discrete rotational concept and to Lagrange-based finite element shell formulations. The competitiveness of the proposed isogeometric shell formulation in terms of computational costs to attain a pre-defined error level is shown. KW - Isogeometric analysis KW - Geometrically nonlinear Reissner–Mindlin shell KW - NURBS KW - Interpolation of rotations KW - Numerical integration Y1 - 2016 U6 - https://doi.org/10.1016/j.cma.2016.01.018 SN - 0045-7825 VL - 303 SP - 1 EP - 34 ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Vitucci, Gennaro A1 - Klinkel, Sven T1 - The weak substitution method – An application of the mortar method for patch coupling in NURBS-based isogeometric analysis T2 - International Journal for Numerical Methods in Engineering N2 - In this contribution, a mortar‐type method for the coupling of non‐conforming NURBS (Non‐Uniform Rational B‐spline) surface patches is proposed. The connection of non‐conforming patches with shared degrees of freedom requires mutual refinement, which propagates throughout the whole patch due to the tensor‐product structure of NURBS surfaces. Thus, methods to handle non‐conforming meshes are essential in NURBS‐based isogeometric analysis. The main objective of this work is to provide a simple and efficient way to couple the individual patches of complex geometrical models without altering the variational formulation. The deformations of the interface control points of adjacent patches are interrelated with a master‐slave relation. This relation is established numerically using the weak form of the equality of mutual deformations along the interface. With the help of this relation, the interface degrees of freedom of the slave patch can be condensated out of the system. A natural connection of the patches is attained without additional terms in the weak form. The proposed method is also applicable for nonlinear computations without further measures. Linear and geometrical nonlinear examples show the high accuracy and robustness of the new method. A comparison to reference results and to computations with the Lagrange multiplier method is given. KW - nonlinear isogeometric analysis KW - NURBS KW - domain decomposition KW - weak substitution method KW - multi-patch connection KW - mortar methods Y1 - 2015 U6 - https://doi.org/10.1002/nme.4918 SN - 1097-0207 VL - 103 IS - 3 SP - 205 EP - 234 ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Klinkel, Sven T1 - Treatment of Reissner–Mindlin shells with kinks without the need for drilling rotation stabilization in an isogeometric framework T2 - Computer Methods in Applied Mechanics and Engineering N2 - 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. KW - Isogeometric analysis KW - Geometrical nonlinear Reissner–Mindlin shell KW - NURBS KW - Interpolation of rotations KW - Treatment of kinks KW - Shell not requiring drilling rotation stabilization Y1 - 2014 U6 - https://doi.org/10.1016/j.cma.2014.03.017 SN - 0045-7825 VL - 276 SP - 35 EP - 66 ER - TY - GEN A1 - Schmidt, Simon A1 - Dornisch, Wolfgang A1 - Müller, Ralf T1 - A phase field model for martensitic transformation coupled with the heat equation T2 - GAMM‐Mitteilungen N2 - In order to consider temperature dependency in a phase field model for martensitic transformations a temperature dependent phase separation potential is introduced. The kinematics and the energetic setup underlying the phase transformation are briefly explained. Parameters are identified using molecular dynamics (MD) simulations. The kinetics of the phase field model are in good agreement with those of the MD simulations. Further, the effect of temperature on the microstructure evolution is studied for varying initial austenite contents. KW - Solidifcation KW - phase-field KW - theory of porous media KW - finite element method KW - multi-scale Y1 - 2017 U6 - https://doi.org/10.1002/gamm.201720005 SN - 1522-2608 VL - 40 IS - 2 SP - 138 EP - 153 ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Müller, Ralf T1 - Modeling of electric field-induced magnetization switching in multi- ferroic heterostructures T2 - Proceedings in Applied Mathematics and Mechanics N2 - Multiferroic heterostructures consist of materials with either pronounced ferroelectric or ferromagnetic effect. The combination of both types of material, be it in layers, columns or inclusions, potentially yields a significant magneto‐electric coupling effect even at room temperature. The magnetization in the ferromagnetic material can be controlled by the application of electric fields to the ferroelectric material. In this contribution a linear elastic continuum formulation is coupled with a phase field formulation for the polarization and magnetization in the ferroelectric and the ferromagnetic layer, respectively. The strain transfer at the interface of the layers yields a magneto‐electric coupling effect within the heterostructures. The finite element method is used to discretize the arising differential equations. A numerical example provides a proof of concept for the simulation of the magneto‐electric coupling effect in multiferroic heterostructures. Y1 - 2019 U6 - https://doi.org/10.1002/pamm.201900103 SN - 1617-7061 VL - 19 IS - 1 ER - TY - GEN A1 - Schmidt, Simon A1 - Dornisch, Wolfgang A1 - Müller, Ralf T1 - Martensitic transformation at a crack under mode I and II loading T2 - Proceedings in Applied Mathematics and Mechanics N2 - Metastable austenitic steels can undergo phase transformation. As an allotrope two crystal configurations are of interest: the softer austenitic parent phase and the martensitic phases. Here, the bain orientation relationship leads to distinct orientations for the martensitic variants with a different transformation strain [7]. A phase field approach is used to model the transformation, where a multi‐valued order parameter urn:x-wiley:16177061:media:PAMM201900465:pamm201900465-math-0001 identifies the austenitic parent phase and the martensitic variants. This allows to define bulk and surface energies as regularized functions in terms of the order parameter and its gradient. The kinetics of the martensitic transformation are temperature dependent. Temperatures below an equilibrium temperature favour the growth of the martensitic phase, whereas temperatures above the equilibrium favour the austenitic phase. Approaching the equilibrium temperature slows down the transformation [5]. In this work we consider a static crack under mode I and mode II loading. Y1 - 2019 U6 - https://doi.org/10.1002/pamm.201900465 SN - 1617-7061 VL - 19 IS - 1 PB - WILEY-VCH Verlag GmbH & Co. KGaA CY - Weinheim ER - TY - GEN A1 - Dornisch, Wolfgang T1 - Efficient Integration Schemes in NURBS-based Isogeometric Analysis – Comparison and Application to Shell Elements T2 - VII International Conference on Isogeometric Analysis N2 - The efficiency of the NURBS-based isogeometric Galerkin method highly depends on the chosen numerical integration scheme for the evaluation of the stiffness matrix and the residual vector. In an efficient implementation, the computational costs for the formation of the stiffness matrix scale almost linearly with the total number of integration points. Furthermore, also the accuracy of the solution depends on the chosen integration scheme, since spurious locking effects increase with the number of integration points. Gauss integration is probably the most common approach for numerical integration in NURBS-based IGA, whereby the higher continuity in IGA can be exploited for a reduction of the number of integration points, see [2]. More sophisticated integration schemes have been proposed in [3,4], where a nonlinear equation is solved in order to compute integration weights and quadrature points. Using the minimum possible number of quadrature points, the nonlinear equation aims at integrating the univariate B-spline spaces exactly. Thus, the methods can be considered as optimal. In this contribution, an implementation of the optimal integration method [3] for general NURBS surfaces with non-uniform knot vectors is presented. The integration method is applied to the Poisson problem, two-dimensional plane stress problems and isogeometric shell elements as proposed in [5]. The numerical examples of this study focus on the accuracy of the integration for NURBS surfaces with non-uniform weights. A further very important aspect is the comparison of computational costs between the different methods. The alleviation of locking effects, which comes along with a reduction of the number of integration points, is studied for the shell elements proposed in [5]. Furthermore, a comparison to the locking-free formulation of [6] is given. Y1 - 2019 UR - http://congress.cimne.com/IGA2019/admin/files/fileabstract/a224.pdf UR - http://congress.cimne.com/iga2019/frontal/ProgramPrint.asp?t=todo PB - CIMNE CY - Barcelona ER - TY - GEN A1 - Chasapi, Margarita A1 - Dornisch, Wolfgang A1 - Klinkel, Sven T1 - Coupling of patches for isogeometric analysis of solids in boundary representation T2 - VII International Conference on Isogeometric Analysis N2 - In this contribution we present a coupling approach for non-conforming NURBS patches in the framework of an isogeometric formulation in boundary representation [1]. In order to fit the boundary representation modeling technique in CAD, we follow the idea of the scaled boundary finite element method [2] for the parameterization of the solid. Thus, two-dimensional solid surfaces are partitioned into sections in relation to a central point, the scaling center. Each section is parameterized with a circumferential parameter along the boundary and a radial scaling parameter in the interior of the domain. We employ NURBS basis functions for the approximation of the solution in both parametric directions. The approximation in scaling direction is flexible and allows for local refinement within the computational domain. For the coupling of non-conforming sections we employ a mortar approach as presented in [3]. We establish a master-slave relation for the interface control points between adjacent sections based on the equality of mutual deformations along the interface. The coupling is realized in a weak manner by constraining the basis functions, which means that the NURBS basis functions of the slave side are related to those of the master side so that the solution of both sections along the interface is equal in a weak sense. This approach is derived numerically and its application to nonlinear problems is straightforward. We study several numerical examples of linear and nonlinear problems in solid mechanics and compare the results with conforming computations. Y1 - 2019 UR - http://congress.cimne.com/IGA2019/admin/files/fileabstract/a184.pdf PB - CIMNE CY - Barcelona ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Schrade, David A1 - Wolf, Jana A1 - Müller, Ralf T1 - Numerical methods for the modeling of the magnetization vector in multiferroic heterostructures T2 - Proceedings in Applied Mathematics and Mechanics, Special Issue: 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Weimar 2017) N2 - Multiferroic heterostructures are commonly used to obtain electro‐magnetic coupling effects. Thereby, the ferroelectric layer is used to control the magnetization in the ferromagnetic layer. The coupling between the layers is obtained by the mechanical coupling between the layers, which have well‐defined interfaces. Within this contribution we use phase field models to define the polarization and magnetization in the ferroelectric and ferromagnetic layers, respectively. A coupling between polarization/magnetization and strains in each layer in combination with coherent deformations at the interface yields an electromagnetic coupling within the entire heterostructure. Numerical formulations for the interpolation of the polarization vector are well‐defined in the literature. However, the establishment of a consistent numerical formulation for the ferromagnetic layer, where the length of the magnetization vector has to be constant, remains a difficult task. We propose a new numerical approach for the consistent treatment of the ferromagnetic layer and provide numerical simulations which illustrate the electromagnetic coupling effect. Y1 - 2017 U6 - https://doi.org/10.1002/pamm.201710221 SN - 1617-7061 VL - 17 IS - 1 SP - 503 EP - 504 PB - WILEY-VCH Verlag GmbH & Co. KGaA CY - Weinheim ER - TY - GEN A1 - Dornisch, Wolfgang A1 - Müller, Ralf ED - Schröder, Jörg ED - Lupascu, Doru C. ED - Wende, Heiko ED - Brands, Dominik T1 - Phase field modelling of ferroelectric-ferromagnetic interfaces T2 - Proceedings of the Third Seminar on The Mechanics of Multifunctional Materials : Physikzentrum Bad Honnef June 11 - 15, 2018 N2 - Magneto-electric coupling along interfaces between ferroelectric and ferromagneticmaterials allows the magnetization in ferromagnetic layers to be controlled by electrical fields.The coupling effect is mainly due to the deformation coupling between the layers. Thus, furthercoupling effects at the interfaces are neglected in this contribution. Phase field formulationsare used to model the polarization and magnetization in the ferroelectric and ferromagneticlayers, respectively. A coupling between the phase field and the strains is introduced in eachlayer in combination with a mechanical coupling at the interface. The numerical formulationfor the ferroelectric layer is taken from literature. A special focus is set on the discretization ofthe length-constrained order parameter in the ferromagnetic layer. We show a new approachto enforce this constraint and provide a numerical simulation which illustrates the magneto-electric coupling effect. Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:hbz:464-20180702-085433-3 UR - https://duepublico.uni-duisburg-essen.de/servlets/DocumentServlet?id=46481 SN - 3-9818074-4-8 SN - 978-3-9818074-4-8 SP - 93 EP - 96 PB - Universität Duisburg-Essen CY - Essen ER - 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 - Held, Susanne A1 - Dornisch, Wolfgang T1 - Mass lumping scheme for IGA dynamics T2 - 11th European Solid Mechanics Conference N2 - The aim of this study is to provide a new mass lumping scheme for explicit dynamic calculations in isogeometric analysis (IGA). Shape functions for IGA methods are taken from the Computer-Aided design (CAD) model. The use of higher-order polynomials, usually Non-Uniform Rational B-Splines (NURBS), ensures an exact geometry description [1]. Thus, in comparison to standard finite element method (FEM) the number of elements and therefore the computational costs can be lowered. In addition to that the convergence rate raises equally with the polynomial order. Increasing the degree of shape functions, less elements are necessary for results of the same quality. In explicit dynamics, mass lumping schemes are commonly applied to reduce computational costs by using diagonal mass matrices. Well-known techniques like row-sum lumping or diagonal scaling method were developed for dynamic analysis with standard FEM. Unfortunately, they are neither suitable for higher-order shape functions in IGA, nor in the spectral element method [2]. It is not possible to take advantage of increasing the polynomial order to lower the number of elements in explicit dynamic IGA, because the error caused by lumping the mass matrix also increases. Thus, other mass lumping schemes, suitable for higher-order shape functions, have to be developed. As they are already used for isogeometric mortar method [3], several types of dual basis functions (duals) are studied for a new mass lumping scheme. Using duals as test functions in IGA leads to diagonal consistent mass matrices [4] or banded matrices, depending on the chosen type of duals. In case of the banded matrices, additional row-sum lumping causes smaller errors than lumping the original mass matrix with standard NURBS as test functions. For IGA dynamics with explicit time integration, e.g. the Central Difference Method, using duals as test functions is a very promising approach. They can be easily implemented in existing methods through multiplying the already assembled system matrices with a transformation matrix based on the underlying NURBS curve. Applying the dual lumping scheme, an explicit dynamic analysis could also be performed efficiently using IGA methods with higher-order shape functions. KW - Mass lumping KW - dual basis functions KW - explicit dynamics Y1 - 2022 UR - https://az659834.vo.msecnd.net/eventsairwesteuprod/production-abbey-public/c1273be32ff3490d890d81b5114e25f8 CY - Galway, Irland ER - TY - GEN A1 - Held, Susanne A1 - Dornisch, Wolfgang ED - Korobenko, Artem ED - Evans, John ED - Hsu, Ming-Chen T1 - Mass lumping with dual test functions in IGA dynamics T2 - Isogeometric Analysis 2022 (IGA 2022) - Book of Abstracts N2 - The focus of this study is on providing an efficient and highly accurate mass lumping scheme for explicit dynamic calculations in isogeometric analysis (IGA). Computer-Aided design models make use of higher-order polynomials, usually Non-Uniform Rational Splines (NURBS), to build up the geometry model. In IGA they are chosen as shape functions to take advantage of their high continuity and to keep the description of geometry exact. Thus, compared to standard Finite Element Method (FEM), a smaller number of elements is required to gain results of the same quality level. In general, explicit time integration methods require a huge number of time steps to obtain numerical stable results of reasonable quality. Hence, diagonal mass matrices are preferred to reduce the computational costs within each time step. Row-sum lumping and diagonal scaling method are two well-known techniques, which have been developed as mass lumping schemes for standard FEM. Unfortunately, these standard FEM mass lumping schemes deteriorate convergence rates in IGA dynamics significantly. With rising order of the basis functions, accuracy of the results decreases. The reason for this is that shape functions of higher order lead to more dense mass matrices and thus the error caused by lumping the mass matrices increases. Therefore, new mass lumping schemes have to be developed, which are more suitable for the use of higher-order shape functions. Several types of dual basis functions (duals), which have already been used in the isogeometric mortar method, are considered for a new mass lumping scheme. Keeping the initial NURBS as shape functions, the duals are applied as test functions for the variational form. Depending on the chosen type of duals, diagonal or banded mass matrices are obtained without any additional lumping. Compared to lumping the original mass matrices based on NURBS as test functions, applying additional row-sum lumping to the banded matrices causes significantly smaller errors. The presented new mass lumping scheme is a very promising approach, in which duals are used as test functions in IGA dynamics with explicit time integration, e.g. the Central Differences Method. The implementation into existing codes can be easily done. The assembled system matrices, if based on NURBS as shape and test functions, only have to be multiplied with a transformation matrix. Numerical examples show that if using this dual lumping scheme within IGA methods, explicit dynamic analysis can be performed efficiently with higher-order shape functions. Y1 - 2022 UR - https://drive.google.com/file/d/1CYA-jTeVREnLcY2XIY1HuYiYutElGxXE/view SP - 44 ER - TY - GEN A1 - Held, Susanne A1 - Dornisch, Wolfgang ED - Klinkel, Sven ED - Klarmann, Simon T1 - Lumping für explizite Zeitintegration in der isogeometrischen Analyse. T2 - Forschungskolloquium 2021 in 2022 : Baustatik-Baupraxis : Kloster Steinfeld Y1 - 2022 SN - 978-3-946090-15-1 SP - 20 EP - 21 PB - RWTH Aachen, Lehrstuhl für Baustatik und Baudynamik CY - Aachen 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 - Held, Susanne A1 - Dornisch, Wolfgang A1 - Azizi, Nima ED - Soriano, Enrique Nadal ED - Cardiel, Carmen Rodrigo ED - Casas, José Martinez T1 - An isogeometric element formulation for linear two-dimensional elasticity based on the Airy equation T2 - Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference N2 - 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. KW - Airy equation KW - Isogeometric Analysis KW - two-dimensional elasticity KW - NURBS KW - element formulation Y1 - 2022 UR - https://yic2021.upv.es/book-of-abstracts/ SN - 978-84-9048-969-7 SP - 118 PB - Editorial Universitat Politècnica de València CY - Valencia, Spanien 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 - 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 - Stammen, Lisa A1 - Dornisch, Wolfgang ED - Klinkel, Sven ED - Klarmann, Simon T1 - Eine konvektive isogeometrische Elementformulierung mit angepassten Interpolationsordnungen T2 - Forschungskolloquium 2021 in 2022 : Baustatik-Baupraxis : Kloster Steinfeld Y1 - 2022 SN - 978-3-946090-15-1 SP - 51 EP - 52 PB - RWTH Aachen, Lehrstuhl für Baustatik und Baudynamik CY - Aachen ER - TY - GEN A1 - Stammen, Lisa A1 - Dornisch, Wolfgang T1 - Investigations on the use of adapted approximation orders for a convective IGA formulation T2 - 11th European Solid Mechanics Conference N2 - In isogeometric analysis (IGA), which was founded by Hughes et al. [1], the geometry representation is used for the analysis as well. Hence, due to the exact description of the geometry, analysis results can be improved [1, 2]. Therefore, different kinds of splines, like non-uniform rational B-splines (NURBS) [3], are used as shape functions for the discretizations. In linear elasticity problems, for standard formulations, shear locking phenomena can occur due to the different orders of the derivatives in the unbalanced strain-displacement relation. This effect can be reduced using shape functions of higher order, causing an increased computational effort. For low-order formulations, the degrees of shape functions can be adapted accordingly in order to counteract this effect. For an isogeometric displacement-stress mixed Reissner-Mindlin shell formulation such adapted approximation spaces were investigated in [4], for instance. In [5], additionally to a proper choice of shape function spaces, convective coordinates are employed in the derivation of isogeometric shell formulations. In this contribution, a convective displacement-based isogeometric formulation is introduced, wherein the displacements in the different surface directions are approximated independently using appropriate approximation orders. Therefore, two different meshes are generated from the NURBSdescribed geometry representation, employing order elevation solely for one of the two surface directions. Thus, the order is elevated in opposite directions for these two meshes. The two different possibilities of order elevation are investigated. Furthermore, the use of different convective basis systems is studied. This comprises convective basis systems in each control point, computed according to [6], as well as convective basis systems determined from the local geometry direction in each integration point. The achieved results are compared to whose of a two-dimensional displacementstress mixed formulation presented in [7]. KW - Isogeometric analysis KW - adapted approximation orders KW - convective basis systems Y1 - 2022 UR - https://az659834.vo.msecnd.net/eventsairwesteuprod/production-abbey-public/cab20a58733c4d95bc33e1b330b064a2 CY - Galway, Irland ER - TY - GEN A1 - Stammen, Lisa A1 - Dornisch, Wolfgang ED - Soriano, Enrique Nadal ED - Cardiel, Carmen Rodrigo ED - Casas, José Martinez T1 - A mixed isogeometric plane stress and plane strain formulation with different continuities for the alleviation of locking T2 - Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference N2 - Isogeometric analysis and mixed finite element methods offer promising opportunities to enhance analysis results for complex problems like incompressible elasticity and are able to cope with different locking phenomena. In this contribution, a mixed two-field isogeometric formulation with independent approximations for displacements and stresses is derived, and its ability to counteract different types of locking is investigated using two examples. Furthermore, the influence of the continuity of the stress shape functions on the accurancy of results and convergence behaviour is shown. KW - Isogeometric Analysis KW - Mixed Formulations KW - Spline Basis Functions KW - Continuity KW - Locking Y1 - 2022 UR - https://yic2021.upv.es/book-of-abstracts/ SN - 978-84-9048-969-7 SP - 114 PB - Editorial Universitat Politècnica de València, CY - Valencia, Spanien 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 -