@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. BeirĖœao 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} } @misc{ChasapiDornischKlinkel, author = {Chasapi, Margarita and Dornisch, Wolfgang and Klinkel, Sven}, title = {Patch coupling in isogeometric analysis of solids inboundary representation using a mortar approach}, series = {International Journal for Numerical Methods in Engineering}, volume = {121}, journal = {International Journal for Numerical Methods in Engineering}, number = {14}, issn = {1097-0207}, doi = {10.1002/nme.6354}, pages = {3206 -- 3226}, abstract = {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.}, 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} } @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{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{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{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{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{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{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{ChasapiDornischKlinkel, author = {Chasapi, Margarita and Dornisch, Wolfgang and Klinkel, Sven}, title = {Coupling of patches for isogeometric analysis of solids in boundary representation}, series = {VII International Conference on Isogeometric Analysis}, journal = {VII International Conference on Isogeometric Analysis}, publisher = {CIMNE}, address = {Barcelona}, pages = {1}, abstract = {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.}, language = {en} }