Isogeometric Reissner Mindlin Shell Research Articles

Mixed dimensional isogeometric FE-BE coupling analysis for solid–shell structures

In this paper, the collocation-based isogeometric boundary element method (IGABEM) is combined with the isogeometric Reissner–Mindlin shell elements to conduct the mixed dimensional solid–shell coupling analysis. On the basis of taking advantages of the geometric smoothness and exactness of isogeometric analysis (IGA), this method can reduce the computational scale of the solid part and eliminate the procedure of constructing analysis-suitable volumetric discretizations. In addition, this method only needs to provide the surface CAD (computer-aided design) meshes of the whole coupling model, so it has great potential in achieving close link with CAD systems. For the coupling implementation, the stiffness formula of BE subdomains is formed by condensing the unknown tractions, which simplifies the implementation of coupling constraints. That is to say, only the displacement compatibility equations on the interface are required, without explicitly establishing the traction equilibrium conditions. Two coupling approaches are developed and studied. One is the collocation-based direct strict kinematic coupling based on the deformation hypothesis of the shell across the section, and another is the weak coupling formulation in an integral sense based on work equality. The accuracy and convergence properties of the presented coupling method are investigated through numerical examples including a complex impeller blade.

Isogeometric symmetric FE-BE coupling method for acoustic-structural interaction

The problems of Acoustic-Structural Interaction (ASI) are commonly encountered in the simulation of the thin-walled structures immersed in the fluid. This interaction can affect the acoustic properties of the fluid and the dynamic characteristics of the structure. In this paper, a Non-Uniform Rational B-Splines (NURBS)-based Isogeometric Finite Element (FE) - Boundary Element (BE) symmetric coupling method is developed to study the ASI problems and the free vibration of the elastic structures submerged in the fluid. The geometry of the thin-walled structures is exactly modelled and discretized by utilizing the Isogeometric Reissner-Mindlin shell elements. This shell theory simplifies the continuity enforcement between NURBS patch boundaries. The Isogeometric Boundary Element Method (IGABEM) is applied to describe the acoustic field. In order to impose coupling constraints, a transformation equation of unknown variables on the interface between sound field and structure field is established, that is, sound pressure versus external force and particle velocity versus displacement. Different from the non-symmetric matrix obtained by the traditional direct BEM, a new variational formulation is introduced to obtain the symmetric coupling coefficient matrix, which maks this coupling approach suitable for a broad class of solvers and improves the robustness of the computation. The reliability and stability of the symmetric coupling method are verified by numerical examples considering acoustic and structural loadings. It is concluded that the symmetric coupling method possesses high accuracy, and can circumvent the limitation of traditional coupling method in solving open boundary problems.

A 3D isogeometric FE-IBE coupling method for acoustic-structural interaction problems with complex coupling models

In this paper, Finite Element-Indirect Boundary Element (FE-IBE) coupling approach based on the Isogeometric Analysis (IGA) framework is proposed to deal with Acoustic-Structural Interaction (ASI) problems. Numerical simulation of thin-walled structures is carried out by using isogeometric Reissner-Mindlin shell theory. The isogeometric indirect boundary element method (IBEM) is used to simulate the fluid domain, and the coupling formula is established by connecting the primary variables. IBEM contains information on both sides of the boundary. Naturally, FE-IBE coupling method inherits these attractive features. FE-IBE coupling method has a wide application prospect. In general, it can not only deal with the common problems, but also solve the problems that Finite Element-Direct Boundary Element (FE-DBE) coupling method cannot handle, such as the opened boundary problems, whose inner and outer boundaries contact the liquid. In order to further expand its application range, based on the universality of the primary variables in IBEM, a more generalized coupling formula is derived to solve the hybrid problems with both the closed and opened boundaries. Numerical examples with analytical solution verify the reliability and stability of the proposed IGA FE-IBE coupling method, and two complex problems in aerospace engineering and ocean engineering are studied in detail.

An isogeometric Reissner–Mindlin shell element based on Bézier dual basis functions: Overcoming locking and improved coarse mesh accuracy

We develop a mixed geometrically nonlinear isogeometric Reissner–Mindlin shell element for the analysis of thin-walled structures that leverages Bézier dual basis functions to address both shear and membrane locking and to improve the quality of computed stresses. The accuracy of computed solutions over coarse meshes, that have highly non-interpolatory control meshes, is achieved through the application of a continuous rotational approach. The starting point of the formulation is the modified Hellinger–Reissner variational principle with independent displacement, membrane, and shear strains as the unknown fields. To overcome locking, the strain variables are interpolated with lower-order spline bases while the variations of the strain variables are interpolated with the corresponding Bézier dual bases. Leveraging the orthogonality property of the Bézier dual basis, the strain variables are condensed out of the system with only a slight increase in the bandwidth of the resulting linear system. The condensed approach preserves the accuracy of the non-condensed mixed approach but with fewer degrees of freedom. From a practical point of view, since the Bézier dual basis is completely specified through Bézier extraction, any spline space that admits Bézier extraction can utilize the proposed approach directly.

Isogeometric FE-BE coupling approach for structural-acoustic interaction

This paper introduces two NURBS (Non-Uniform Rational B-Splines)-based Isogeometric FE (Finite Element)-BE (Boundary Element) coupling approaches, i.e. DCM (Directed Coupling Method) and SICM (Symmetric-Iterative Coupling Method), to solve the structural-acoustic interaction problems. IGA (IsoGeometric Analysis) can eliminate the substantial geometrical errors and time-consuming meshing steps. Meanwhile, CAD (Computer Aided Design) only makes use of surface representations to model geometrical objects, which is naturally combined with BEM (Boundary Element Method). Therefore, the IGABEM (Isogeometric Boundary Element Method) simulates the acoustic domain described by the Helmholtz equation, while structures are discretized by using the Isogeometric Reissner-Mindlin shell elements that only require C0-continuity across the element boundaries, which simplifies the continuity enforcement between patch boundaries. The main strategy of two coupling approaches is to establish the transformation of interface boundary conditions, that is, the sound pressure is converted to the force boundary conditions of shell structure. The reliability and accuracy of DCM are verified by analytical solutions or Lagrange-based coupling method. By using the model with six identical bi-quartic NURBS patches, the geometric singularity of the spherical surface is eliminated, and the accuracy is significantly improved. It is found that the interaction between the fluid and structure is closely related to the physical properties of the fluid. Compared with Lagrange model, IGA method has much higher accuracy and convergence. SICM method is introduced for the first time to deal with the structure-sound interaction problem. Although SICM fails near the natural frequency of the structure, it has the advantage of generating symmetric coupling matrix, which greatly reduces the dimension of the matrix. Finally, a complex toroidal shell subjected to the mixed structural-acoustic loads, including the plane acoustic wave and the concentrated load, is investigated.

Adjusted approximation spaces for the treatment of transverse shear locking in isogeometric Reissner–Mindlin shell analysis

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.

An isogeometric Reissner–Mindlin shell element based on mixed grid

We propose in this article a new isogeometric Reissner–Mindlin degenerated shell element for linear analysis. It is based on the mixed use of non-uniform rational basis spline and Lagrange basis functions in the same domain. The mid-surface of the shell is represented and discretized using non-uniform rational basis spline and the directors of the shell are discretized using Lagrange polynomials. The interpolatory property of Lagrange polynomials gives a natural choice of fiber vectors, thus removing the difficulties in the definition of directors that is seen in most isogeometric Reissner–Mindlin shell elements. The non-uniform rational basis spline representation of the mid-surface allows us to maintain the exact geometry representation characteristic of the isogeometric approach. The independent expressions of displacements and rotations also give users the possibility to use different numbers of degrees of freedom in an element for both kinematic variables. Several numerical examples show that our method is simple, robust, and efficient.

Implicit dynamic analysis using an isogeometric Reissner–Mindlin shell formulation

SummaryIn 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. Copyright © 2016 John Wiley & Sons, Ltd.

An isogeometric Reissner-Mindlin shell element for dynamic analysis considering geometric and material nonlinearities

The present approach deals with the dynamical analysis of thin structures using an isogeometric Reissner-Mindlin shell formulation. Here, a consistent and a lumped mass matrix are employed for the implicit time integration method. The formulation allows for large displacements and finite rotations. The Rodrigues formula, which incorporates the axial vector is used for the rotational description. It necessitates an interpolation of the director vector in the current configuration. Two concept for the interpolation of the director vector are presented. They are denoted as continuous interpolation method and discrete interpolation method. The shell formulation is based on the assumption of zero stress in thickness direction. In the present formulation an interface to 3D nonlinear material laws is used. It leads to an iterative procedure at each integration point. Here, a J2 plasticity material law is implemented. The suitability of the developed shell formulation for natural frequency analysis is demonstrated in numerical examples. Transient problems undergoing large deformations in combination with nonlinear material behavior are analyzed. The effectiveness, robustness and superior accuracy of the two interpolation methods of the shell director vector are investigated and are compared to numerical reference solutions.

Reissner–Mindlin shell implementation and energy conserving isogeometric multi‐patch coupling

SummaryA shear‐flexible isogeometric Reissner–Mindlin shell formulation using non‐uniform rational B‐splines basis functions is introduced, which is used for the demonstration of a coupling approach for multiple non‐conforming patches. The six degrees of freedom formulation uses the exact surface normal vectors and curvature. The shell formulation is implemented in an isogeometric analysis framework for computation of structures composed of multiple geometric entities. To enable local model refinement as well as non‐matching domains coupling, a conservative multi‐patch approach using Lagrange multipliers for structured non‐uniform rational B‐splines patches is presented. Here, an additional border frame mesh is used to couple geometries during structural analyses. This frame interface approach avoids the problem of excessive constraints when multiple patches are coupled at one point. First, the shell formulation is verified with several reference cases. Then the influence of the frame interface discretization and frame penalty stiffness on the smoothness of the results is investigated. The effects of the perturbed Lagrangian method in combination with the frame interface approach is shown. In addition, results of models with T‐joint interface connections and perpendicular stiffener patches are presented. Copyright © 2016 John Wiley & Sons, Ltd.

An efficient and robust rotational formulation for isogeometric Reissner–Mindlin shell elements

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.

An efficient and robust Reissner–Mindlin shell formulation for isogeometric analysis

AbstractThe novel idea of isogeometric analysis is to use the basis functions of the geometry description of the design model also for the analysis. Thus, the geometry is represented exactly on element level. A closer integration of design and analysis is fostered by the usage of one common geometry model for design and analysis. A prevalent choice for the geometry description in isogeometric shell analysis are Non‐Uniform Rational B‐spline (NURBS) surfaces, which are commonly used in industrial design software to model thin structures. In order to directly compute structures defined by NURBS surfaces, an efficient isogeometric shell formulation is required. In this contribution an isogeometric Reissner–Mindlin shell formulation derived from the continuum theory is presented. The shell body is described by a shell reference surface, which is defined by NURBS surfaces, and a director vector. The director vector in the current configuration is computed by an orthogonal rotation using Rodrigues' tensor in every integration point. The axial vector of the rotation is interpolated. A multiplicative update formulation for the rotations accounts for finite rotations. A benchmark example shows the superior accuracy of the presented shell formulation for nonlinear computations. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Developments of the mixed grid isogeometric Reissner–Mindlin shell: Serendipity basis and modified reduced quadrature

In this paper, we use the Serendipity basis in the isogeometric Reissner–Mindlin shell formulation to facilitate the fiber vector definition. The Serendipity basis and the NURBS basis are used here to express the shell fiber rotations and the mid-surface translations respectively. Results show that they perform nearly the same with the Lagrange/NURBS formulation, but with savings in the number of rotational degrees of freedom. We also study the transverse shear locking phenomenon and give a modified reduced quadrature scheme. It can improve the efficiency and to some extent relieve the locking without introducing hourglass modes. The linear decreasing, from outer elements to inner elements, quadrature point distribution also prevents the oscillation of the solutions in the coarse meshes.

Treatment of Reissner–Mindlin shells with kinks without the need for drilling rotation stabilization in an isogeometric framework

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.

Isogeometric Reissner–Mindlin shell analysis with exactly calculated director vectors

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.