Isogeometric Kirchhoff Love Shell Research Articles

Goal-adaptive Meshing of Isogeometric Kirchhoff–Love Shells

AbstractMesh adaptivity is a technique to provide detail in numerical solutions without the need to refine the mesh over the whole domain. Mesh adaptivity in isogeometric analysis can be driven by Truncated Hierarchical B-splines (THB-splines) which add degrees of freedom locally based on finer B-spline bases. Labeling of elements for refinement is typically done using residual-based error estimators. In this paper, an adaptive meshing workflow for isogeometric Kirchhoff–Love shell analysis is developed. This framework includes THB-splines, mesh admissibility for combined refinement and coarsening and the Dual-Weighted Residual (DWR) method for computing element-wise error contributions. The DWR can be used in several structural analysis problems, allowing the user to specify a goal quantity of interest which is used to mark elements and refine the mesh. This goal functional can involve, for example, displacements, stresses, eigenfrequencies etc. The proposed framework is evaluated through a set of different benchmark problems, including modal analysis, buckling analysis and non-linear snap-through and bifurcation problems, showing high accuracy of the DWR estimator and efficient allocation of degrees of freedom for advanced shell computations.

Multi-patch isogeometric Kirchhoff–Love shell analysis for post-buckling of functionally graded graphene platelets reinforced composite shells

This paper develops a multi-patch isogeometric Kirchhoff–Love shell method for post-buckling of functionally graded graphene platelets reinforced composite (FG-GPLRC) cylindrical, spherical, and conoidal shell structures, which are built with single or multiple NURBS patches. A penalty strategy is employed to weakly couple nonconforming interfaces between adjacent patches. The coupling work induced by enforcing displacement continuity and rotational continuity is added to the equilibrium equation, and the corresponding stiffness matrix is derived in detail. A simplified arc-length method is utilized to capture the complex equilibrium paths including snap-through and snap-back behaviors. Five distribution patterns of the shells including uniform (UD), V-type, A-type, O-type, and X-type are considered. The cylindrical and spherical shells are subjected to concentrated loadings at central points while for conoidal shells the concentrated loadings are enforced at the center points of an edge. The post-buckling of isotropic and laminated shell structures is first studied to validate the developed formulations by comparing the obtained results with those given in existing literature. Then a series of numerical examples considering nonlinear FG-GPLRC shell problems are conducted to explore the effect of various parameters like geometric dimensions, GPL distribution patterns, and shell thickness on the mechanical performance. Finally, the post-buckling of a cylindrical shell subjected to an offset concentrated load, with extremely complicated equilibrium paths, is modeled and analyzed by using the developed multi-patch isogeometric method. The numerical results reveal that the X-type GPL distribution pattern demonstrates better performance in load–deflection responses and provides the largest buckling critical load among the five patterns. Additionally, the increase in height ratio could deteriorate the stability performance of FG-GPLRC conoidal shells.

Isogeometric Dynamic Buckling Analysis of Trimmed and Multipatch Thin-Shell Structures

This paper extends our previous work on the isogeometric dynamic buckling analysis of thin-shell structures to the trimmed and multipatch situation where features such as cutouts and stiffeners can be easily incorporated. To be specific, a modified generalized- time integration scheme combined with a geometric nonlinear isogeometric Kirchhoff–Love shell element is used to simulate the complex buckling and postbuckling behaviors of thin-shell structures. The developed method can damp properly high-frequency contents while maintaining second-order accuracy in the dynamic buckling analysis. For the integration of arbitrary-shaped trimmed elements, a geometrically exact blending function method is developed to improve the efficiency of the dynamic shell buckling analysis. To deal with multipatch geometries, a penalty-based weak coupling approach is developed, where coupled patches with nonconforming trimmed interfaces or even with prescribed angles, such as stiffeners, can be analyzed. We demonstrate the accuracy, stability, and flexibility of the proposed framework with several numerical examples. In particular, the influences of “free” and “partially free” control points, penalty factor, trimming, as well as different modeling strategies on the dynamic solutions of shell structures are investigated.

Isogeometric analysis for multi-patch structured Kirchhoff–Love shells

We present an isogeometric method for Kirchhoff–Love shell analysis of shell structures with geometries composed of multiple patches and which possibly possess extraordinary vertices, i.e. vertices with a valency different to four. The proposed isogeometric shell discretisation is based on the one hand on the approximation of the mid-surface by a particular class of multi-patch surfaces, called analysis-suitable G1 (Collin et al., 2016), and on the other hand on the use of the globally C1-smooth isogeometric multi-patch spline space (Farahat et al., 2023). We use our developed technique within an isogeometric Kirchhoff–Love shell formulation (Kiendl et al., 2009) to study linear and non-linear shell problems on multi-patch structures. Thereby, the numerical results show the great potential of our method for efficient shell analysis of geometrically complex multi-patch structures which cannot be modelled without the use of extraordinary vertices.

Large deformation frictional contact formulations for isogeometric Kirchhoff–Love shell

The large deformation frictional contact between shell structures is commonplace in engineering applications like the sheet metal forming process. Due to the specific assumptions of shell structures, the numerical calculation of shell contact is more complicated than the computation of contact between normal two or three dimensional solids. In recent years, the Kirchhoff–Love shell has become one of the most popular mathematical shell models under the context of isogeometric analysis. In this paper, we detailedly formulate the frictional contact contributions for Kirchhoff–Love shell structures considering both stick and slide states. The variation and linearization of contact variables considered to be sophisticated are explicitly derived and plainly organized. The general nonlinear hyperelastic materials are utilized to describe the constitutive relations of shell structures undergoing large deformation. The normal and tangential contact constraints are enforced using the classical penalty method. Based on the precise representation of shell structures using NURBS functions, we developed an effective contact detection algorithm to accurately capture the contact areas between shells. A computation framework for solving Kirchhoff–Love shell frictional contact problems is established in the context of NURBS-based isogeometric analysis. Several shell contact examples are calculated, and the obtained results are compared with those of the classical finite element method. The convergence and accuracy of the presented method are verified in these examples.

Implicit dynamic buckling analysis of thin‐shell isogeometric structures considering geometric imperfections

AbstractIn continuation of our previous work on nonlinear stability analysis of trimmed isogeometric thin shells, this contribution is an extension to dynamic buckling analyses for predicting reliably complex snap‐through and mode‐jumping behavior. Specifically, a modified generalized‐ time integration scheme is used and combined with a nonlinear isogeometric Kirchhoff–Love shell element to provide second‐order accuracy while introducing controllable high‐frequency dissipation. In addition, a weak enforcement of essential boundary conditions based on a penalty approach is considered with a particular focus on the inhomogeneous case of imposed prescribed displacements. Moreover, we propose a least‐squares B‐spline surface fitting approach and corresponding error measures to model both eigenmode based and measured geometric imperfections. The imperfect geometries thus obtained can be naturally integrated into the framework of isogeometric nonlinear dynamic shell analysis. Based on this idea, the different modeling methods and the influence of the appropriately considered geometric imperfections on the dynamic buckling behavior can be investigated systematically. Both perfect and geometrically imperfect shell models are considered to assess the performance of the proposed method. We compare our method with established developments in this field and demonstrate superior achievements with regard to solution quality and robustness.

Penalty coupling of trimmed isogeometric Kirchhoff–Love shell patches

Abstract We present a formulation for isogeometric Kirchhoff–Love shell analysis on complex CAD models consisting of multiple trimmed patches. The method is based on the penalty coupling method presented in Herrema AJ, Johnson EL, Proserpio D, Wu MCH, Kiendl J, Hsu MC (Penalty coupling of non-matching isogeometric Kirchhoff–Love shell patches with application to composite wind turbine blades. Computer Methods in Applied Mechanics and Engineering 2019;346:810–840.) and extended to the application on arbitrary coupling curves defined either in the physical or parametric space. We present the detailed formulation ready for implementation. Different numerical tests demonstrate the accuracy and applicability of the method.

Isogeometric Kirchhoff–Love shell patches in free and forced vibration of sinusoidally corrugated FG carbon nanotube-reinforced composite panels

Application of the isogeometric method is illustrated for free and forced vibration analyzes of sinusoidally corrugated functionally graded carbon nanotube reinforced composite (FG-CNTRC) thin panels. These panels can be used as a potential structure for morphing skin applications. The nanocomposite panel is fabricated from a polymeric matrix which is reinforced by carbon nanotubes (CNTs). The volume fraction of CNTs varies based on the four models of functionally graded (FG). The Kirchhoff–Love (KL) shell theory is employed to model the kinematic behavior of sinusoidally corrugated panels (SCPs) with a variable radius of curvature. The multiple patches approach is approved to model the structure. In this model, the bending strip method (BSM) is applied at the intersection of the patches. The temporal response of the panel under different transverse dynamical loadings is predicted via the Newmark time-marching scheme. A detailed comparison study is performed to demonstrate the validity and efficiency of the proposed isogeometric analysis (IGA). The numerical results demonstrate that the computationally effective IGA can provide accurate results due to the capability of precisely approximating any segments of the corrugation. Furthermore, the effects of different configurations of CNTs and various geometrical parameters such as corrugation amplitude, number of corrugation units and panel aspect ratio are presented on the natural frequency and dynamic response of the sinusoidally corrugated FG-CNTRC panels.

Phase-field simulation of ductile fracture in shell structures

In this paper, a computational framework for simulating ductile fracture in multipatch shell structures is presented. A ductile fracture phase-field model at finite strains is combined with an isogeometric Kirchhoff–Love shell formulation. For the application to complex structures, we employ a penalty approach for imposing, at patch interfaces, displacement and rotational continuity and C0 and C1 continuity of the phase-field, the latter required if a higher-order phase-field formulation is adopted. We study the mesh dependency of the numerical model and we show that mesh refinement allows for capturing important features of ductile fracture such as cracking along shear bands. Therefore, we investigate the effectiveness of a predictor–corrector algorithm for adaptive mesh refinement based on LR NURBS. Thanks to the adoption of time- and space-adaptivity strategies, it is possible to simulate the failure of complex structures with a reasonable computational effort. Finally, we compare the predictions of the numerical model with experimental results.

An updated Lagrangian framework for Isogeometric Kirchhoff–Love thin-shell analysis

We propose a comprehensive Isogeometric Kirchhoff–Love shell framework that is capable of undergoing large elasto-plastic deformations. Central to this development, we reformulate the governing thin-shell equations in terms of the mid-surface velocity degrees of freedom, accommodating the material response in the time-rate form while ensuring objectivity. To handle complex multipatch geometries, we propose a consistent penalty coupling technique for enforcing the continuity conditions at patch interfaces. Penalty is also employed to weakly enforce symmetry boundary conditions. A recently proposed non-local penalty contact is adopted as part of the formulation in order to handle complex dynamic crushing simulations. Numerical examples, ranging from static elasto-plastic shell benchmarks to highly dynamic crushing scenarios, validate the accuracy, efficiency and robustness of the proposed framework.

A framework for efficient isogeometric computations of phase-field brittle fracture in multipatch shell structures

We present a computational framework for applying the phase-field approach to brittle fracture efficiently to complex shell structures. The momentum and phase-field equations are solved in a staggered scheme using isogeometric Kirchhoff–Love shell analysis for the structural part and isogeometric second- and fourth-order phase-field formulations for the brittle fracture part. For the application to complex multipatch structures, we propose penalty formulations for imposing all the required interface constraints, i.e., displacement (C0) and rotational (C1) continuity for the structure as well as C0 and C1 continuity for the phase field, where the latter is required only in the case of the fourth-order phase-field model. All involved penalty terms are scaled with the corresponding problem parameters to ensure a consistent scaling of the penalty contributions to the global system of equations. As a consequence, all coupling terms are controlled by one global penalty parameter, which can be set to 103 independent of the problem parameters. Furthermore, we present a multistep predictor–corrector algorithm for adaptive local refinement with LR NURBS, which can accurately predict and refine the region around the crack even in cases where fracture fully develops in a single load step, such that rather coarse initial meshes can be used, which is essential especially for the application to large structures. Finally, we investigate and compare the numerical efficiency of loosely vs. strongly staggered solution schemes and of the second- vs. fourth-order phase-field models.

A robust penalty coupling of non-matching isogeometric Kirchhoff–Love shell patches in large deformations

Isogeometric Kirchhoff–Love elements have been receiving increasing attention in geometrically nonlinear analysis of thin shells because they make it possible to meet the C1 requirement in the interior of surface patches and to avoid the use of finite rotations. However, engineering structures of appreciable complexity are typically modeled using multiple patches and, often, neither rotational continuity nor conforming discretization can be practically obtained at patch interfaces. Simple penalty approaches for coupling adjacent patches, applicable to either smooth or non-smooth interfaces and either matching or non-matching discretizations, have been proposed. Although the problem dependence of the penalty coefficient can be reduced by scaling factors which take into account geometrical and material parameters, only high values of the penalty coefficient can guarantee a negligible coupling error in all possible cases. However, this can lead to an ill conditioned problem and to an increasing iterative effort for solving the nonlinear discrete equations. In this work, we show how to avoid this drawback by rewriting the penalty terms in an Hellinger–Reissner form, introducing independent fields work-conjugated to the coupling equations. This technique avoids convergence problems, making the analysis robust also for very high values of the penalty coefficient, which can be then employed to avert coupling errors. Moreover, a proper choice of the basis functions for the new fields provides an accurate coupling also for general non-matching cases, preventing overconstrained solutions. The additional variables are condensed out and then not involved in the global system of equations to be solved. A highly efficient approach based on a mixed integration point strategy and an interface-wise reduced integration rule makes the condensation inexpensive preserving the sparsity of the condensed stiffness matrix and the coupling accuracy.

An nth high order perturbation-based stochastic isogeometric method and implementation for quantifying geometric uncertainty in shell structures

This paper presents an n-th high order perturbation-based stochastic isogeometric Kirchhoff–Love shell method, formulation and implementation for modeling and quantifying geometric (thickness) uncertainty in thin shell structures. Firstly, the Non-Uniform Rational B-Splines (NURBS) is used to describe the geometry and interpolate the variables in a deterministic aspect. Then, the shell structures with geometric (thickness) uncertainty are investigated by developing an nth order perturbation-based stochastic isogeometric method. Here, we develop the shell stochastic formulations in detail (particularly, expand the random input (thickness) and IGA Kirchhoff-Love shell element based state functions analytically around their expectations via n-th order Taylor series using a small perturbation parameterε), whilst freshly providing the Matlab core codes helpful for implementation. This work includes three key novelties: 1) by increasing/utilizing the high order of NURBS basis functions, we can exactly represent shell geometries and alleviate shear locking, as well as providing more accurate deterministic solution hence enhancing stochastic response accuracy. 2) Via increasing the nth order perturbation, we overcome the inherent drawbacks of first and second-order perturbation approaches, and hence can handle uncertainty problems with some large coefficients of variation. 3) The numerical examples, including two benchmarks and one engineering application (B-pillar in automobile), simulated by the proposed formulations and direct Monte Carlo simulations (MCS) verify that thickness randomness does strongly affect the response of shell structures, such as the displacement caused by uncertainty can increase up to 35%; Moreover, the proposed formulation is effective and significantly efficient. For example, compared to MCS, only 0.014% computational time is needed to obtain the stochastic response.

The embedded isogeometric Kirchhoff–Love shell: From design to shape optimization of non-conforming stiffened multipatch structures

Isogeometric shape optimization uses a unique model for the geometric description and for the analysis. The benefits are multiple: in particular, it avoids tedious procedures related to mesh updates. However, although the analysis of complex multipatch structures now becomes tractable with advanced numerical tools, isogeometric shape optimization has not yet been proven to be applicable for designing such structures. Based on the initial concept of integrating design and analysis, we develop a new approach that deals with the shape optimization of non-conforming multipatch structures. The model is built by employing the Free-Form Deformation principle. Introducing NURBS composition drastically simplifies the imposition of the shape updates in case of a non-conforming multipatch configuration. In the case of stiffened structures, the use of embedded surfaces enables to tackle the geometric constraint of connecting interfaces between the panel and the stiffeners during shape modifications. For the analysis, we introduce the embedded Kirchhoff–Love shell formulation. The NURBS composition defines the geometry of the shell while the displacement field is approximated using the same spline functions as for the embedded surface. We also formulate a new mortar method to couple non-conforming Kirchhoff–Love shells which intersect with any angle. We apply the developed method on different examples to demonstrate its efficiency and its potential to optimize complex industrial structures in a smooth manner.

Penalty coupling of non-matching isogeometric Kirchhoff–Love shell patches with application to composite wind turbine blades

Isogeometric analysis (IGA) has been a particularly impactful development in the realm of Kirchhoff–Love thin-shell analysis because the high-order basis functions employed naturally satisfy the requirement of C1 continuity. Still, engineering models of appreciable complexity, such as wind turbine blades, are typically modeled using multiple surface patches and, often, neither rotational continuity nor conforming discretization can be practically obtained at patch interfaces. A penalty approach for coupling adjacent patches is therefore presented. The proposed method imposes both displacement and rotational continuity and is applicable to either smooth or non-smooth interfaces and either matching or non-matching discretization. The penalty formulations require only a single, dimensionless penalty coefficient for both displacement and rotation coupling terms, alleviating the problem-dependent nature of the penalty parameters. Using this coupling methodology, numerous benchmark problems encapsulating a variety of analysis types, geometrical and material properties, and matching and non-matching interfaces are addressed. The coupling methodology produces consistently accurate results throughout all tests. Furthermore, the suggested penalty coefficient of α=103 is shown to be effective for the wide range of problem configurations addressed. Finally, a realistic wind turbine blade model, consisting of 27 patches and 51 coupling interfaces and having a chordwise- and spanwise-variant composite material definition, is subjected to buckling, vibration, and nonlinear deformation analyses using the proposed approach.

Shape Optimization for Natural Frequency with Isogeometric Kirchhoff-Love Shell and Sensitivity Mapping

A fast shape optimization strategy for free form shell structure design with structural dynamics criteria is proposed in this paper. The structures are modelled with Non-Uniform Rational B-Spline based isogeometric Kirchhoff-Love shell elements. The substitution of the traditional finite elements not only makes the mesh model geometrically exact but also avoids the laborious mesh regeneration during the design update. As for the structural response evaluation, the modal synthesis method is adopted to avoid a repeated evaluation of some substructures where there are no designed variables attached; thus, the model reanalysis is speeded up. A bottom-up strategy for the analytical design sensitivity evaluation is also proposed here; the element-level analytical sensitivity with respect to the inherent shape parameters is firstly calculated from which the design sensitivity is then extracted with the help of a sensitivity map. Finally, gradient based algorithm is used to solve the optimization problem. Several examples show that our approach is flexible and efficient for fast free form shell structure optimization.

A flexible approach for coupling NURBS patches in rotationless isogeometric analysis of Kirchhoff–Love shells

This paper presents a flexible method for coupling NURBS patches in isogeometric Kirchhoff–Love shell analysis. The required C1-continuity in such a shell formulation significantly complicates the patch coupling (as compared to typical C0-cases). In the present work, the C0-part of the coupling is a global coupling in a weak sense, whereas the C1-continuity is enforced by a strong point-wise coupling in well-chosen collocation points along the interface. The coupling conditions can be derived using only mesh information, without the need for suitable penalty or stabilisation parameters. They are expressed using a master–slave formulation between the interface variables. A static condensation approach to enforce these continuity constraints results in a reduced system matrix. The proposed method can be employed for both conforming and non-conforming patch configurations, and for G1-continuous structures as well as for patches meeting at a kink. This is demonstrated for a set of problems of (dynamic) shell analysis, including both eigenvalue and boundary-value problems.

Modal Synthesis with the Isogeometric Kirchhoff–Love Shell Elements

The modal synthesis method is frequently used for the analysis of large structures composed of multiple parts concerning dynamic aspects. In this paper, we extended the modal synthesis method under the isogeometric analysis framework. The isogeometric Kirchhoff–Love shell elements are used for the analysis of the substructures, the Craig–Bampton method is used for the modal synthesis and the bending-strip method is used for the substructures coupling. We give examples on the modal analysis and the harmonic response analysis. The results show the effectiveness of the method.