Reissner-Mindlin Shell Elements Research Articles

A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition

A curved non-isoparametric Reissner–Mindlin shell element is developed for analyzing thin-walled structures. The standard kinematic description of the element requires the calculation of the director vector. To address this demand accurately, similar to isogeometric analysis (IGA), the geometry is defined by utilization of the non-uniform rational B-splines (NURBS) imported directly from computer-aided design (CAD) files. Then, shape functions of the Legendre spectral element method (SEM) are used to interpolate the displacements. Consequently, the shell director vector and Jacobian of the transformation are calculated properly according to the presented formulation. On the other hand, in Legendre SEM combined with Gauss–Lobatto–Legendre quadrature, the integration points and the element nodes coincide. Thus, the easily computable local coordinate systems at the integration points can be used directly as nodal basis systems. A separate calculation of nodal basis systems at control points, which is the source of either complexity or error in IGA shells, is not required. Given the condition number of the stiffness matrix in the developed method, super high-order elements can also be used. Very high order p-refined elements are used in addition to h-refinement of the mesh to show the capability of higher order elements to analyze problems without mesh refinement. The validity and convergence rate of the method are investigated and verified through various cases of h- and p-refinement in challenging obstacle course problems.

Transverse shear parametrization in hierarchic large rotation shell formulations

AbstractConsistent treatment of large rotations in common Reissner–Mindlin formulations is a complicated task. Reissner–Mindlin formulations that use a hierarchic parametrization provide an elegant way to facilitate large rotation shell analyses. This can be achieved by the assumption of linearized transverse shear strains, resulting in an additive split of strain components, which technically simplifies implementation of corresponding shell finite elements. The present study aims at validating this assumption by systematically comparing numerical solutions with those of a newly implemented hierarchic and fully nonlinear Reissner–Mindlin shell element.

A non-local approach for Reissner–Mindlin shell elements in dynamic simulations: Application with a Gurson model

Numerical prediction of ductile tearing during a car crash is necessary to ensure the structural integrity of the vehicle. Recently, encouraging results were obtained by Davaze et al., (2021) with a non-local damage approach, compatible with dynamic explicit simulations and preserving parallel computation: an implicit gradient model enabled to reproduce experimental results with DP450 steel sheets with 3D solid elements. However, in the automotive industry, thin sheets of metals are mostly modeled with shell elements for computation time efficiency. Shell elements have a more complex formulation: their kinematic description is enriched accounting for displacements but also rotations, leading to two different strain contributions: membrane strain and bending strain. The transverse strain is computed independently in a post-processing step from local internal variables with a zero normal stress assumption. This formulation is known to lead to strain localization after the onset of necking, while this is not the case with 3D solid elements. The problem is even more challenging when considering shells in association with a softening material accounting for damage as this also leads to strong mesh dependency, as with 3D solid elements. This complex problem is solved using non-local formulations for the material behavior. The proposed solution accounts for both in-plane and in-thickness damage gradients. It consists of a two-step regularization procedure introducing nested wire elements in the shell thickness. Using non-local variables to evaluate the transverse strain enables to obtain a continuous spatial distribution for strains, in agreement with the 3D solid case. An updated Lagrangian formulation is used to account for large strains. The performance of the strategy is demonstrated in three test cases, with a comparison with an experimental result for one of them. The results obtained following the proposed strategy lead are equivalent to the ones obtained with the non-local method proposed by Davaze et al., (2021) and are thus consistent with the experimental data.

Translational surface coupling along a line with non-conforming meshes

In the modelling of large-scale metal structures, comprising plated components intersecting along weld lines, the meshing efficiency and flexibility are limited by the requirements of nodal alignment and compliance of element size, shape and edge orientation throughout the domain. Such limitations necessitate the use of complex transitional meshes in intersection regions and result in highly complex global mesh configurations. This paper presents an original and systematic methodology for surface coupling along an arbitrary 1D interface, which is applicable to any type of 2D and 3D FEs, and which provides a systematic framework for: (i) geometric modelling of weld lines; (ii) coupling of regions with different levels of discretisation detail or element types within a system; and (iii) domain partitioning problems involving computationally heterogeneous partitions. The strategy is based upon a novel coupling element formulation, which uses the fundamental principles of the mortar method and an augmented Lagrangian Multiplier optimisation approach. Particular consideration is given to an element formulation that enforces rigid translational coupling, which has been implemented for employment with co-rotational Reissner-Mindlin shell elements. Various numerical examples are presented to demonstrate the accuracy, versatility and substantial computational benefits of the developed methodology for modelling large-scale metal structural systems.

Isogeometric analysis with the Reissner-Mindlin shell for hull structural mechanical analysis

Isogeometric Analysis (IGA) is an important emerging numerical method that is being studied in various field. IGA is proposed to seamlessly connect the Computer-Aided Engineering (CAE) and the Computer-Aided Design (CAD) systems, so that the costly mesh generating stage in traditional Finite Element Method (FEM) can be skipped to improve the efficiency of design and analysis. In this paper, an IGA approach is introduced to the hull structural mechanical analysis. A Reissner-Mindlin shell element is constructed in the IGA framework, and the expression of the element stiffness matrix and load vector are derived in detail. Using the proposed method, a typical hull block structure is analyzed under both bending and torsion load. The maximum relative error of the displacement results is 1.6% compared to the reference results obtained from ANSYS 17.0, and the correctness of the proposed method is verified. It also proves that the proposed method can be used in engineering practice, and can be used to realize the integration of design and analysis.

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.

Galerkin formulations of isogeometric shell analysis: Alleviating locking with Greville quadratures and higher-order elements

We propose new quadrature schemes that asymptotically require only four in-plane points for Reissner–Mindlin shell elements and nine in-plane points for Kirchhoff–Love shell elements in B-spline and NURBS-based isogeometric shell analysis, independent of the polynomial degree p of the elements. The quadrature points are Greville abscissae associated with pth-order B-spline basis functions whose continuities depend on the specific Galerkin formulations, and the quadrature weights are calculated by solving a linear moment fitting problem in each parametric direction. The proposed shell element formulations are shown through numerical studies to be rank sufficient and to be free of spurious modes. The studies reveal comparable accuracy, in terms of both displacement and stress, compared with fully integrated spline-based shell elements, while at the same time reducing storage and computational cost associated with forming element stiffness and mass matrices and force vectors. The high accuracy with low computational cost makes the proposed quadratures along with higher-order spline bases, in particular polynomial orders, p=5 and 6, good choices for alleviating membrane and shear locking in shells.

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.

Torsional and Transversal Stiffness of Orthotropic Sandwich Panels.

In the present work, an analytical equation describing the plate torsion test taking into account the transverse shear stiffness in sandwich plates is derived and numerically validated. Transverse shear becomes an important component if the analyzed plate or shell is thick with respect to the in-plane dimensions and/or its core has significantly lower stiffness than the outer faces. The popular example of such a sandwich plate is a corrugated cardboard, widely used in the packaging industry. The flat layers of a corrugated board are usually made of thicker (stronger) material than that used for the corrugated layer, the role of which is rather to keep the outer layers at a certain distance, to ensure high bending stiffness of the plate. However, the soft core of such a plate usually has a low transverse shear stiffness, which is often not considered in the plate analysis. Such simplification may lead to significant calculation errors. The paper presents the generalization of the Reissner’s analytical formula, which describes the torsional stiffness of the plate sample including two transverse shear stiffnesses. The paper also presents the implementation of the numerical model of the plate torsion test including the transverse shear stiffnesses. Both approaches are compared with each other on a wide range of material parameters and different aspect ratios of the specimen. It has been proved that both analytical and numerical formulations lead to an identical result. Finally, the performance of presented formulations is compared with other numerical models using commercial implementation of various Reissner–Mindlin shell elements and other analytical formulas from the literature. The comparison shows good agreement of presented theory and numerical implementation with other existing approaches.

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.

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.

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.

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.

Nonlinear Analysis of RC Structures using Assumed Strain RM Shell Element

Nonlinear analysis of reinforced concrete structures is carried out by using Reissner-Mindlin (RM) shell finite element (FE). The brittle inelastic characteristic of concrete material is represented by using the elasto-plastic fracture (EPF) material model with the relevant material models such as cracking criteria, shear transfer model and tension stiffening model. In particular, assumed strains are introduced in the formulation of the present shell FE in order to avoid element deficiencies inherited in the standard RM shell FE. The arc-length control method is used to trace the full load-displacement path of reinforced concrete structures. Finally, four benchmark tests are carried out and numerical results are provided as future reference solutions produced by RM shell element with assumed strains.

FREE-VIBRATION ANALYSIS OF PLATES AND SHELLS WITH A NINE-NODE ASSUMED NATURAL DEGENERATED SHELL ELEMENT

A study on the free-vibration analysis of plates and shells is described in this paper. In order to determine the natural frequencies of plates and shells, a nine-node degenerated shell element is developed by using the degenerated solid concept based on Reissner–Mindlin (RM) assumptions which allow the shear deformation and rotatory effect to be considered. All terms required in the shell finite element (FE) formulation are defined in the natural domain. In particular, assumed natural strains are derived to alleviate the locking phenomena inherited in the RM shell elements. The natural constitutive equation is used in conjunction with the natural strain terms. The proposed shell FE formulation offers significant implementation advantages since it consistently uses the natural co-ordinate system. Various numerical examples are carried out and its results are then compared with the existing exact solutions and the numerical solutions calculated by other shell FEs.

Shell topology optimization using the layered artificial material model

A general methodology for topology optimization using the finite element method is described for shell structures. Four- and nine-node Reissner–Mindlin shell elements with drilling degrees of freedom are used for the finite element response analysis. The artificial material model is used in the topology optimization and in particular, an isotropic multi-layer shell model is introduced to allow the formation of holes or stiffening zones. In addition, a single design variable resizing algorithm is implemented based on the existing criterion which is found to be adequate for the artificial material model. Several benchmark tests are presented to show the overall performance of the proposed methodology. The strain energy variation together with the variation of the layout of the structure is monitored. Some detailed examples are provided with comparisons of the use of the four- and nine-node elements and studies of critical solution parameters. Copyright 2000 John Wiley & Sons, Ltd.