In this paper we present an isogeometric formulation for rotation-free thin shell analysis of structures comprised of multiple patches. The structural patches are
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1- or higher-order continuous in ...the interior, and are joined with
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0-continuity. The Kirchhoff–Love shell theory that relies on higher-order continuity of the basis functions is employed in the patch interior as presented in Kiendl et al. 36. For the treatment of patch boundaries, a method is developed in which strips of fictitious material with unidirectional bending stiffness and zero membrane stiffness are added at patch interfaces. The direction of bending stiffness is chosen to be transverse to the patch interface. This choice leads to an approximate satisfaction of the appropriate kinematic constraints at patch interfaces without introducing additional stiffness to the shell structure. The attractive features of the method include simplicity of implementation and direct applicability to complex, multi-patch shell structures. The good performance of the bending strip method is demonstrated on a set of benchmark examples. Application to a wind turbine rotor subjected to realistic wind loads is also shown. Extension of the bending strip approach to the coupling of solids and shells is proposed and demonstrated numerically.
A new concept called analysis in computer aided design (AiCAD) is proposed for design-through-analysis workflow. This concept uses non-uniform rational B-Splines (NURBS)-based B-Rep models for the ...entire workflow. Such models consist of trimmed NURBS surfaces and are considered standard in the industry, especially for modeling free-form geometries. The newly developed isogeometric B-Rep analysis (IBRA) used in AiCAD is also presented. IBRA can be considered as a generalization of isogeometric analysis (IGA) that uses the boundary representation (B-Rep) of the design model in addition to the same basis functions as in IGA for approximating the solution fields. IBRA provides the framework for creating a direct and complete analysis model from computer aided design (CAD) in a consistent finite-element-like manner. Thus, IBRA allows analyzing a CAD model without remodeling and meshing, even for complex geometries.
For the numerical integration of trimmed surfaces, the concept of nested Jacobian approach (NEJA) with NURBS surfaces is introduced. In addition, for enforcing the different types of boundary conditions or mechanical entities, a new finite element type called isogeometric B-Rep element is introduced. Elements of this type permit enforcing, e.g., coupling or Dirichlet boundary conditions. A corresponding formulation based on a penalty approach is presented as well.
The proposed workflow is realized exemplarily for surface modeling and the geometrical nonlinear analysis of shell structures. The differences between the standard analysis procedure and the AiCAD workflow are explained in detail. Various numerical examples confirm the accuracy, flexibility, and robustness of the proposed IBRA concept, thus highlighting its advantages for the realization of design-through-analysis workflow with a uniform geometry representation.
•The concept of isogeometric B-Rep analysis (IBRA) is introduced as extension of IGA.•IBRA is based on original CAD models (i.e. trimmed NUBRS based B-Rep models).•A new B-Rep element formulation to couple trimmed multi-patch shells is introduced.•The basis for an integrated design-through-analysis workflow is elaborated in detail.•Various benchmarks show the potential of IBRA (robustness, flexibility and accuracy)
A Kirchhoff–Love shell element is developed on the basis of the isogeometric approach
16. NURBS as basis functions for analysis have proven to be very efficient and offer the great feature of exact ...geometric representation. For a Kirchhoff–Love shell element they additionally have the significant advantage that the necessary continuities between elements are easily achieved. The element is formulated geometrically nonlinear. It is discretized by displacement degrees of freedom only. Aspects related to rotational degrees of freedom are handled by the displacement control variables, too. A NURBS-based CAD program is used to model shell structures built up from NURBS and isogeometric analysis is performed on the same model without meshing. Different examples show the performance of this method and its applicability for the integration of design and analysis.
We present isogeometric shape optimization for shell structures applying sensitivity weighting and semi-analytical analysis. We use a rotation-free shell formulation and all involved geometry models, ...i.e., initial design, analysis model, optimization model, and final design use the same geometric basis, in particular NURBS. A sensitivity weighting scheme is presented which eliminates certain effects of the chosen discretization on the design update. A multilevel design approach is applied such that the design space can be chosen independently from the analysis space. The use of semi-analytical sensitivities allows having different polynomial degrees for design and analysis model. Different numerical examples are performed which confirm the applicability of the proposed method. Furthermore, a shape optimization example with an exact solution is presented which can serve as general benchmark for shape optimization methods.
Nonlinear isogeometric spatial Bernoulli beam Bauer, A.M.; Breitenberger, M.; Philipp, B. ...
Computer methods in applied mechanics and engineering,
05/2016, Letnik:
303
Journal Article
Recenzirano
A new element formulation of a geometrically nonlinear spatial curved beam assuming Bernoulli theory including torsion without warping is proposed. The element formulation is derived directly from ...the 3D-continuum by means of nonlinear kinematics, thus accounting for large displacements. The geometric description of the proposed element is adapted from the spatial rod of Greco and Cuomo (2013) and extended to a nonlinear element formulation. The proposed formulation can handle arbitrary orientations of the cross section along the beam.
In this publication, NURBS are used as basis functions for discretization, since they can easily provide the required C1-continuity between elements. The presented element formulation has four degrees of freedom, three for displacements and one for the rotation around the center line. In order to prove the accuracy of the developed spatial Bernoulli beam, several numerical examples are presented and compared to analytic solutions and other element formulations.
► An isogeometric shell element based on the thin shell theory is proposed. ► The PHT-spline tremendously facilitates local refinement and possess
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1 continuity. ► The thin shell analysis based on ...Kirchhoff–Love theory avoids the use of rotational degrees of freedom.
This paper presents a novel approach for isogeometric analysis of thin shells using polynomial splines over hierarchical T-meshes (PHT-splines). The method exploits the flexibility of T-meshes for local refinement. The main advantage of the PHT-splines in the context of thin shell theory is that it achieves
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1 continuity, so the Kirchhoff–Love theory can be used in pristine form. No rotational degrees of freedom are needed. Numerical results show the excellent performance of the present method.
The penalty method has proven to be a well-suited approach for the application of coupling and boundary conditions on (trimmed) multi-patch NURBS shell structures within isogeometric analysis. Beside ...its favorable simplicity and efficiency, the main challenge is the appropriate choice of the underlying penalty factor — choosing the penalty factor too low yields a poor constraint accuracy, while choosing it too high causes numerical issues like ill-conditioned system matrices or a small infeasible time step size in explicit dynamics. Although recommendations for penalty values exist, profound methods for its determination are still an active field of research.
We address this issue and provide formulas allowing an a priori determination of penalty factors for NURBS-based Reissner–Mindlin shells with penalty-based coupling and boundary conditions. The underlying approach is inspired by a methodology previously used for conventional finite elements, for which penalty factors are derived through a comparison with exact Lagrange multiplier solutions. In that way, penalty formulas consisting of a problem-dependent factor and a problem-independent intensity factor are obtained. The fact that the latter is a direct measure of the penalty-induced error is the main advantage of this approach and enables a problem-independent definition of the penalty factor as a function of the desired accuracy.
We demonstrate the validity of the derived formulas and the corresponding error measure with benchmark problems in linear elasticity including trimmed non-matching NURBS shells. Furthermore we show that the mesh-adaptivity of the penalty formulas improves the convergence behavior and conditioning of penalty methods.
•We present penalty determination formulas for isogeometric Reissner–Mindlin shells.•The formulas distinguish in-plane and out-of-plane displacement penalty factors.•A priori estimation of the introduced penalty error via a penalty intensity factor.•Applicability to arbitrarily trimmed patches with Dirichlet and coupling conditions.•Mesh-adaptive penalty factor to improve convergence and conditioning.
The core idea of this article is nested parametrization in the context of isogeometric analysis. The method has been inspired by trimming procedures and can be applied to different applications like ...local modifications and enhancements of thin-walled structures or coupling of two overlying elements by embedding one in the other.
A remodeling for an explicit representation of the boundaries is avoided, which would be contradictory to the aim of isogeometric analysis of using the original CAD model. The nested entity is directly linked to the super element, however its element formulation is independent of the super element formulation within this article. The derivation, implementation and application is shown in particular for one-dimensional entities embedded into 2D domains. Consequently, the definition of an embedded curve in the surface is required and realized by using NURBS curves in the parameter space of the corresponding surface. This curve with its respective predefined base vectors serves as the basis for new element formulations or adaptation of already developed element formulations, which are based on the geometric description of a curve.
In detail, an adapted formulation of the recently developed nonlinear isogeometric spatial Bernoulli Beam by the authors is presented in this paper. Furthermore, those embedded curves are used for line supports and loads, as well as a mass manipulation. The accuracy of the proposed element formulation is verified by several benchmark examples and the potential for future applications is briefly revealed.
•A general concept for embedding in NURBS-based isogeometric analysis is developed.•The presented technique permits structural entities in higher-dimensional geometries.•Embedded nonlinear isogeometric spatial Bernoulli beam element formulation.•Examples for embedding structural elements, mass and boundary conditions.
