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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 -