@misc{SchmidtDornischMueller, author = {Schmidt, Simon and Dornisch, Wolfgang and M{\"u}ller, Ralf}, title = {A phase field model for martensitic transformation coupled with the heat equation}, series = {GAMM-Mitteilungen}, volume = {40}, journal = {GAMM-Mitteilungen}, number = {2}, issn = {1522-2608}, doi = {10.1002/gamm.201720005}, pages = {138 -- 153}, abstract = {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.}, language = {en} } @misc{ZouScottBordenetal., author = {Zou, Zhihui and Scott, Michael A. and Borden, Michael J. and Thomas, Derek C. and Dornisch, Wolfgang and Brivadis, Ericka}, title = {Isogeometric B{\´e}zier dual mortaring: Refineable higher-order spline dual bases and weakly continuous geometry}, series = {Computer Methods in Applied Mechanics and Engineering}, volume = {333}, journal = {Computer Methods in Applied Mechanics and Engineering}, issn = {0045-7825}, doi = {10.1016/j.cma.2018.01.023}, pages = {497 -- 534}, abstract = {In this paper we develop the isogeometric B{\´e}zier dual mortar method. It is based on B{\´e}zier extraction and projection and is applicable to any spline space which can be represented in B{\´e}zier 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 B{\´e}zier dual mortaring. Weakly continuous geometry is a geometry description where the weak continuity constraints are built into properly modified B{\´e}zier 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.}, language = {en} } @misc{ChenKlinkelDornisch, author = {Chen, Lin and Klinkel, Sven and Dornisch, Wolfgang}, title = {Hybrid collocation-Galerkin approach for the analysis of surface represented 3D-solids employing SB-FEM}, series = {Computer Methods in Applied Mechanics and Engineering}, volume = {295}, journal = {Computer Methods in Applied Mechanics and Engineering}, issn = {0045-7825}, doi = {10.1016/j.cma.2015.07.004}, pages = {268 -- 289}, abstract = {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.}, language = {en} } @misc{DornischVitucciKlinkel, author = {Dornisch, Wolfgang and Vitucci, Gennaro and Klinkel, Sven}, title = {The weak substitution method - An application of the mortar method for patch coupling in NURBS-based isogeometric analysis}, series = {International Journal for Numerical Methods in Engineering}, volume = {103}, journal = {International Journal for Numerical Methods in Engineering}, number = {3}, issn = {1097-0207}, doi = {10.1002/nme.4918}, pages = {205 -- 234}, abstract = {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.}, language = {en} } @misc{KlinkelChenDornisch, author = {Klinkel, Sven and Chen, Lin and Dornisch, Wolfgang}, title = {A NURBS based hybrid collocation-Galerkin method for the analysis of boundary represented solids}, series = {Computer Methods in Applied Mechanics and Engineering}, volume = {284}, journal = {Computer Methods in Applied Mechanics and Engineering}, issn = {0045-7825}, doi = {10.1016/j.cma.2014.10.029}, pages = {689 -- 711}, abstract = {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.}, language = {en} } @misc{DornischKlinkel, author = {Dornisch, Wolfgang and Klinkel, Sven}, title = {Treatment of Reissner-Mindlin shells with kinks without the need for drilling rotation stabilization in an isogeometric framework}, series = {Computer Methods in Applied Mechanics and Engineering}, volume = {276}, journal = {Computer Methods in Applied Mechanics and Engineering}, issn = {0045-7825}, doi = {10.1016/j.cma.2014.03.017}, pages = {35 -- 66}, abstract = {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.}, language = {en} } @misc{DornischKlinkelSimeon, author = {Dornisch, Wolfgang and Klinkel, Sven and Simeon, Bernd}, title = {Isogeometric Reissner-Mindlin shell analysis with exactly calculated director vectors}, series = {Computer Methods in Applied Mechanics and Engineering}, volume = {253}, journal = {Computer Methods in Applied Mechanics and Engineering}, issn = {0045-7825}, doi = {10.1016/j.cma.2012.09.010}, pages = {491 -- 504}, abstract = {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.}, language = {en} } @phdthesis{Dornisch, author = {Dornisch, Wolfgang}, title = {Interpolation of Rotations and Coupling of Patches in Isogeometric Reissner- Mindlin Shell Analysis}, publisher = {Schriftenreihe des Lehrstuhls f{\"u}r Baustatik und Baudynamik der RWTH Aachen}, address = {Aachen}, isbn = {978-3-946090-02-1}, pages = {229}, abstract = {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.}, language = {en} } @misc{KikisDornischKlinkel, author = {Kikis, Georgia and Dornisch, Wolfgang and Klinkel, Sven}, title = {Adjusted approximation spaces for the treatment of transverse shear locking in isogeometric Reissner-Mindlin shell analysis}, series = {Computer Methods in Applied Mechanics and Engineering}, volume = {354}, journal = {Computer Methods in Applied Mechanics and Engineering}, issn = {0045-7825}, doi = {10.1016/j.cma.2019.05.037}, pages = {850 -- 870}, abstract = {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.}, language = {en} } @misc{DornischSchradeXuetal., author = {Dornisch, Wolfgang and Schrade, David and Xu, Bai-Xiang and Keip, Marc-Andr{\´e} and M{\"u}ller, Ralf}, title = {Coupled phase field simulations of ferroelectric and ferromagnetic layers in multiferroic heterostructures}, series = {Archive of Applied Mechanics}, volume = {89}, journal = {Archive of Applied Mechanics}, number = {6}, issn = {0939-1533}, doi = {10.1007/s00419-018-1480-9}, pages = {1031 -- 1056}, abstract = {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.}, language = {en} } @misc{NadgirDornischMuelleretal., author = {Nadgir, Omkar and Dornisch, Wolfgang and M{\"u}ller, Ralf and Keip, Marc-Andr{\´e}}, title = {A phase-field model for transversely isotropic ferroelectrics}, series = {Archive of Applied Mechanics}, volume = {89}, journal = {Archive of Applied Mechanics}, number = {6}, issn = {0939-1533}, doi = {10.1007/s00419-019-01543-y}, pages = {1057 -- 1068}, abstract = {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.}, language = {en} } @misc{DornischStoecklerMueller, author = {Dornisch, Wolfgang and St{\"o}ckler, Joachim and M{\"u}ller, Ralf}, title = {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}, series = {Computer Methods in Applied Mechanics and Engineering}, volume = {316}, journal = {Computer Methods in Applied Mechanics and Engineering}, issn = {0045-7825}, doi = {10.1016/j.cma.2016.07.038}, pages = {449 -- 496}, abstract = {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.}, language = {en} } @misc{GravenkampNatarajanDornisch, author = {Gravenkamp, Hauke and Natarajan, Sundarajan and Dornisch, Wolfgang}, title = {On the use of NURBS-based discretizations in the scaled boundary finite element method for wave propagation problems}, series = {Computer Methods in Applied Mechanics and Engineering}, volume = {315}, journal = {Computer Methods in Applied Mechanics and Engineering}, issn = {0045-7825}, doi = {10.1016/j.cma.2016.11.030}, pages = {867 -- 880}, abstract = {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.}, language = {en} } @misc{SobotaDornischMuelleretal., author = {Sobota, Paul M. and Dornisch, Wolfgang and M{\"u}ller, Ralf and Klinkel, Sven}, title = {Implicit dynamic analysis using an iso- geometric Reissner-Mindlin shell formulation}, series = {International Journal for Numerical Methods in Engineering}, volume = {110}, journal = {International Journal for Numerical Methods in Engineering}, number = {9}, issn = {1097-0207}, doi = {10.1002/nme.5429}, pages = {803 -- 825}, abstract = {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.}, language = {en} } @misc{DornischMuellerKlinkel, author = {Dornisch, Wolfgang and M{\"u}ller, Ralf and Klinkel, Sven}, title = {An efficient and robust rotational formulation for isogeometric Reissner-Mindlin shell elements}, series = {Computer Methods in Applied Mechanics and Engineering}, volume = {303}, journal = {Computer Methods in Applied Mechanics and Engineering}, issn = {0045-7825}, doi = {10.1016/j.cma.2016.01.018}, pages = {1 -- 34}, abstract = {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.}, language = {en} } @misc{DornischMueller, author = {Dornisch, Wolfgang and M{\"u}ller, Ralf}, title = {Modeling of electric field-induced magnetization switching in multi- ferroic heterostructures}, series = {Proceedings in Applied Mathematics and Mechanics}, volume = {19}, journal = {Proceedings in Applied Mathematics and Mechanics}, number = {1}, issn = {1617-7061}, doi = {10.1002/pamm.201900103}, pages = {2}, abstract = {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.}, language = {en} } @misc{SchmidtDornischMueller, author = {Schmidt, Simon and Dornisch, Wolfgang and M{\"u}ller, Ralf}, title = {Martensitic transformation at a crack under mode I and II loading}, series = {Proceedings in Applied Mathematics and Mechanics}, volume = {19}, journal = {Proceedings in Applied Mathematics and Mechanics}, number = {1}, publisher = {WILEY-VCH Verlag GmbH \& Co. KGaA}, address = {Weinheim}, issn = {1617-7061}, doi = {10.1002/pamm.201900465}, pages = {2}, abstract = {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.}, language = {en} } @misc{DornischStoecklerMueller, author = {Dornisch, Wolfgang and St{\"o}ckler, Joachim and M{\"u}ller, Ralf}, title = {Recent advances in isogeometric dual mortar patch coupling}, series = {Proceedings of the 7th GACM Colloquium on Computational Mechanics for Young Scientists from Academia and Industry, October 11-13, 2017 in Stuttgart, Germany}, journal = {Proceedings of the 7th GACM Colloquium on Computational Mechanics for Young Scientists from Academia and Industry, October 11-13, 2017 in Stuttgart, Germany}, publisher = {Institute for Structural Mechanics University of Stuttgart}, address = {Stuttgart}, doi = {10.18419/opus-9334}, url = {http://nbn-resolving.de/urn:nbn:de:bsz:93-opus-ds-93516}, pages = {467 -- 470}, abstract = {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.}, language = {en} } @misc{KikisDornischKlinkel, author = {Kikis, Georgia and Dornisch, Wolfgang and Klinkel, Sven}, title = {A method for the elimination of shear locking effects in an isogeometric Reissner-Mindlin shell formulation}, series = {Proceedings of the 7th GACM Colloquium on Computational Mechanics for Young Scientists from Academia and Industry, October 11-13, 2017 in Stuttgart, Germany}, journal = {Proceedings of the 7th GACM Colloquium on Computational Mechanics for Young Scientists from Academia and Industry, October 11-13, 2017 in Stuttgart, Germany}, publisher = {Institute for Structural Mechanics University of Stuttgart}, address = {Stuttgart}, doi = {10.18419/opus-9334}, pages = {500 -- 503}, abstract = {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.}, language = {en} } @misc{KikisDornischKlinkel, author = {Kikis, Georgia and Dornisch, Wolfgang and Klinkel, Sven}, title = {Isogeometric Reissner-Mindlin shell analysis - adjusted approximation spaces for the reduction of shear locking effects}, series = {VI International Conference on Isogeometric Analysis}, journal = {VI International Conference on Isogeometric Analysis}, publisher = {CIMNE}, address = {Barcelona}, pages = {1}, abstract = {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.}, language = {en} }