Reissner Mindlin Shell Research Articles

The Immersed Boundary Conformal Method for Kirchhoff–Love and Reissner–Mindlin shells

This work utilizes the Immersed Boundary Conformal Method (IBCM) to analyze linear elastic Kirchhoff–Love and Reissner–Mindlin shell structures within an immersed domain framework. Immersed boundary methods involve embedding complex geometries within a background grid, which allows for great flexibility in modeling intricate shapes and features despite the simplicity of the approach. The IBCM method introduces additional layers conformal to the boundaries, allowing for the strong imposition of Dirichlet boundary conditions and facilitating local refinement. In this study, the construction of boundary layers is combined with high-degree spline-based approximation spaces to further increase efficiency. The Nitsche method, employing non-symmetric average operators, is used to couple the boundary layers with the inner patch, while stabilizing the formulation with minimal penalty parameters. High-order quadrature rules are applied for integration over cut elements and patch interfaces. Numerical experiments demonstrate the efficiency and accuracy of the proposed formulation, highlighting its potential for complex shell structures modeled through Kirchhoff–Love and Reissner–Mindlin theories. These tests include the generation of conformal interfaces, the coupling of Kirchhoff–Love and Reissner–Mindlin theories, and the simulation of a cylindrical shell with a through-the-thickness crack.

A three-dimensional meshless fluid–shell interaction framework based on smoothed particle hydrodynamics coupled with semi-meshless thin shell

Meshless methods are suitable for fluid–structure interaction simulations due to its Lagrangian feature and capability of handling large deformations. A three-dimensional meshless framework for the fluid–structure interaction simulation with shell structures is proposed in this study. The weakly compressible smoothed particle hydrodynamics is deployed in the fluid domain, where the boundary integral terms are incorporated to handle the one-layer shell boundary. On the other hand, a semi-meshless finite volume modeling method based on the Reissner–Mindlin shell is deployed in the one-layer shell domain. A one-way coupling method is employed for the interactive forcing between the fluid and shell particles near the interface, where the boundary integral method is employed to approximate the gradient and Laplacian operators in the fluid governing equations. Regarding the no-slip boundary, the original boundary integral method always leads to the artificial slipping phenomenon between the fluid and shell interface. To this end, a velocity penalty term with a relaxation factor is proposed to enforce the no-slip boundary condition, which substantially improves the accuracy of the original boundary integral method. In order to reduce computational costs, a multi-resolution strategy for the meshless framework is employed. A set of challenging cases with low and high Reynolds numbers are simulated by the present framework, and good accuracy is observed with the velocity penalty term in these cases. Overall, the new framework shows a satisfactory performance.

Isogeometric method for buckling prediction and post-buckling analysis of variable stiffness composite underwater pressure shell

Composite cylindrical pressure hulls are thin-walled structures widely used for autonomous underwater vehicles. Buckling failure is one of the most important failure modes for these shells under external pressure. In existing buckling studies of cylindrical pressure hulls, FEM is the most popular analysis method but not efficient enough when dealing with structures with complex material distributions such as the variable stiffness composite shells. Motivated by this, an isogeometric method for buckling prediction and post-buckling analysis of variable stiffness composite underwater pressure shell is proposed in this article. In this method, based on Reissner–Mindlin shell formulas undergoing large deformation, a buckling analysis framework in IGA forms is established. Then a modified arc-length method is proposed based on kinematics used in the shell formulas, and is used to overcome the inaccuracy caused by the 2 degrees of freedom node rotation in the prediction step of arc-length method. In addition, the influence of bifurcation is considered, which may seriously affect the precision of buckling behavior simulation when the limit point is close to a bifurcation point. The computational accuracy of this framework has been validated in a series of constant stiffness composite shell cases involving buckling load prediction and post-buckling behavior simulation. For variable stiffness composite hull cases, this framework achieves the same precision as FEM with fewer elements. Therefore, the proposed framework provides an efficient way for the buckling design of underwater variable stiffness composite pressure hulls.

Isogeometric Shape Optimization of Reissner–Mindlin Shell with Analytical Sensitivity and Application to Cellular Sandwich Structures

Structural shape optimization plays a significant role in structural design, as it can find an appropriate layout and shape to improve structural performance. Isogeometric analysis provides a promising framework for structural shape optimization, unifying the design model and analysis model in the optimization process. This paper presents an adjoint-based analytical sensitivity for isogeometric shape optimization of Reissner–Mindlin shell structures. The shell structures are modeled by multiple NURBS surfaces and design variables are associated with the position of control points. A multilevel approach is performed with a coarse mesh for the design model and a dense mesh for the analysis model. The sensitivity propagation is achieved through a transformation matrix between the design and analysis models. Structural compliance minimization problems with and without constraints are studied and the optimization history shows that the optimization can converge quickly within fewer iterations. The developed formulations are validated through several numerical examples and applied to the optimization of cellular sandwich structures, which are widely used in engineering applications. Numerical results show that optimized sandwich panels can achieve better performance in bending resistance.

Simultaneous analysis of continuously embedded Reissner–Mindlin shells in 3D bulk domains

AbstractA mechanical model and numerical method for the simultaneous analysis of Reissner–Mindlin shells with geometries implied by a continuous set of level sets (isosurfaces) over some three‐dimensional bulk domain is presented. A three‐dimensional mesh in the bulk domain is used in a tailored FEM formulation where the elements are by no means conforming to the level sets representing the shape of the individual shells. However, the shell geometries are bounded by the intersection curves of the level sets with the boundary of the bulk domain so that the boundaries are meshed conformingly. This results in a method which was coined Bulk Trace FEM before. The simultaneously considered, continuously embedded shells may be useful in the structural design process or for the continuous reinforcement of bulk domains. Numerical results confirm higher‐order convergence rates.

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

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

Phase‐field model for ductile fracture in the stress resultant geometrically exact shell

SummaryWe present a phase‐field fracture model for a stress resultant geometrically exact shell in finite deformation regime where the configuration manifold evolves according to deformation and fracture. The Reissner–Mindlin shell problem is first solved via the finite element method, where the independent unknown fields are the displacement and director. The phase‐field ductile fracture model is then coupled with the verified geometrically exact shell by enriching the three‐field elasto‐plastic free energy with a regularized crack surface energy. To capture the geometrical nonlinearity due to the large plastic deformation exhibited in the processing zone, the corresponding finite‐strain stress resultant elasto‐plasticity model is coupled with the phase field model. This coupling between the stress resultant elasto‐plasticity model and the phase‐field fracture model enables us to predict the different amounts of energy dissipation attributed to plastic deformation and fracture and hence simulate the crack propagation in the ductile regime properly. We introduce a mixed finite element model with displacement, director, and phase‐field order parameters as the nodal degree of freedoms formulated on the mid‐surface. An energy‐based arc‐length method is generalized to track the equilibrium path and mitigate instabilities arising from material nonlinearity. Four numerical examples are presented to validate the implementation of the model and demonstrate its capability to simulate ductile fracture in shell structures. These examples include a plane‐stress tension/shear fracture model, the Muscat‐Fenech and Atkins plate, axial tension of a notched cylindrical shell, and ductile fracture of a simply supported plate under uniform pressure.

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.

Nonlinear combined harmonic resonances of composite cylindrical shells operating in hygro-thermo-electro-magneto-mechanical fields

By considering the effects of the hygro-thermo-electro-magnetic environments, von Karman nonlinear terms, and multi-harmonic excitations, a coupled nonlinear vibration modeling of composite cylindrical shells comprising a carbon nanotube-reinforced composite (CNTRC) core and two piezoelectric/magnetic composite (PEMC) skins is developed, and the nonlinear dynamic behaviors of such cylindrical shells under primary, super/sub-harmonic, and combined resonance states are investigated. In the theoretical modeling process, the effective mechanical properties of the CNTRC core are first determined using the mixing and Schapery laws, and the hygro-thermo-electro-magneto-mechanical constitutive relations of the PEMC skins are then formulated. Within the framework of Reissner-Mindlin shell theory, the Lagrangian of the system containing Green-Lagrange and von Karman nonlinear terms is derived, and solution techniques based on the multiscale method are provided to obtain nonlinear frequencies, dynamic responses, and phases of CNTRC-PEMC cylindrical shells under multi-physics fields. Consequently, comparison studies are conducted to validate the correctness of the proposed model from different aspects. Based on this, various resonance and chaos behaviors of such structures subjected to multiple load components are revealed and compared, and the influences of environmental factors and structure composition on the amplitude-frequency curves, time-history responses, and phase planes are explored, with several recommendations and findings being drawn.

A novel explicit design method for complex thin-walled structures based on embedded solid moving morphable components

In this article, a novel explicit approach for designing complex thin-walled structures based on the Moving Morphable Component (MMC) method is proposed, which provides a unified framework to systematically address various design issues, including topology optimization, reinforced-rib layout optimization, and sandwich structure design problems. The complexity of thin-walled structures mainly comes from flexible geometries and the variation of thickness. On the one hand, the geometric complexity of thin-walled structures leads to the difficulty in automatically describing material distribution (e.g., reinforced ribs). On the other hand, thin-walled structures with different thicknesses require various hypotheses (e.g., Kirchhoff–Love shell theory and Reissner–Mindlin shell theory) to ensure the precision of structural responses. Whereas for cases that do not fit the shell hypothesis, the precision loss of response solutions is non-negligible in the optimization process since the accumulation of errors will cause entirely different designs. Hence, the current article proposes a novel embedded solid component to tackle these challenges. The geometric constraints that make the components fit to the curved thin-walled structure are whereby satisfied. Compared with traditional strategies, the proposed method is free from the limit of shell assumptions of structural analysis and can achieve optimized designs with clear load transmission paths at the cost of few design variables and degrees of freedom for finite element analysis (FEA). Finally, we apply the proposed method to several representative examples to demonstrate its effectiveness, efficiency, versatility, and potential to handle complex industrial structures.

On the compatibility between Reissner–Mindlin shell and GBT-based finite elements

This technical note discusses aspects related to the combination of standard Reissner–Mindlin shell and GBT-based (beam) finite elements, with the purpose of modelling, efficiently (with degree-of-freedom economy) and accurately, the structural behaviour of thin-walled members and frames with complex geometry, undergoing global-local-distortional deformation. Following the approach previously proposed by the authors (Manta et al., 2020 [1]; 2021 [2]; 2020 [3]), each element type is employed where it is most efficient: (i) GBT-based elements in the prismatic zones and (ii) shell elements in the geometrically complex zones (e.g., tapered, joints, etc.). While lack of compatibility obviously leads to over-flexible solutions, it is shown that full compatibility can lead to locking. Consequently, a well-balanced compatibility statement is proposed. Several numerical examples are presented to show the advantages of the proposed approach over the previous one, in linear static and buckling (bifurcation) analyses.

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.

Isogeometric analysis using G-spline surfaces with arbitrary unstructured quadrilateral layout

G-splines are a generalization of B-splines that deals with extraordinary points by imposing G1 constraints across their spoke edges, thus obtaining a continuous tangent plane throughout the surface. Using the isoparametric concept and the Bubnov–Galerkin method to solve partial differential equations with G-splines results in discretizations with global C1 continuity in physical space. Extraordinary points (EPs) are required to represent manifold surfaces with arbitrary topological genus. In this work, we allow both interior and boundary EPs and there are no limitations regarding how close EPs can be from each other. Reaching this level of flexibility is necessary so that splines with EPs can become mainstream in the design-through-analysis cycle of the complex thin-walled structures that appear in engineering applications. To the authors’ knowledge, the two EP constructions based on imposing G1 constraints proposed in this work are the first two EP constructions used in isogeometric analysis (IGA) that combine the following distinctive characteristics: (1) Only vertex-based control points are used and they behave as geometric shape handles, (2) any control point of the control net can potentially be an EP, (3) global C1 continuity in physical space is obtained without introducing singularities, (4) faces around EPs are not split into multiple elements, i.e., Bézier meshes with uniform element size are obtained, and (5) good surface quality is attained. The studies of convergence and surface quality performed in this paper suggest that G-splines are more suitable for IGA than EP constructions based on the D-patch framework. Finally, we have represented the stiffener, the inner part, and the outer part of a B-pillar with G-spline surfaces and solved eigenvalue problems using both Kirchhoff–Love and Reissner–Mindlin shell theories. The results are compared with bilinear quadrilateral meshes and excellent agreement is found between G-splines and conventional finite elements. In summary, G-splines are a viable alternative to design and analyze thin-walled structures using the same geometric representation so as to streamline the design-through-analysis cycle.

An edge center based strain-smoothing element with discrete shear gap for the analysis of Reissner–Mindlin shell

In this paper, an edge center based strain-smoothing element with discrete shear gap method (ECSSE-DSG) based on triangular Reissner–Mindlin flat shell element is proposed to solve shell problems. In the proposed ECSSE, the strains for three edge centers were calculated by the strain-smoothing method, which performed strain-smoothing operation directly on the three edges of a linear finite element instead of requiring an extra smoothing domain. In this way, not only could a linear strain field be constructed within an element, but strain consistency at the center of the element’s intersecting edges could also be ensured. The proposed ECSSE was adopted to manage membrane and bending strains. Meanwhile, the discrete shear gap (DSG) method was organically combined with the ECSSE for processing the shear strain to eliminate shear-locking. The ECSSE-DSG showed outstanding performance in a variety of tests, including those for plates, folded plates and curved shells; static, free vibration, dynamic, and buckling analyses; benchmark and engineering examples. High accuracy was obtained for both displacement and strain energy, there was no appearance of shear-locking, a smoother stress field was produced, and there was temporal stability. The excellent performance of the ECSSE-DSG when employed for engineering problems indicated its promising practical application prospects.

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.

Static Analysis of Stiffened Shells Using an Edge-Based Smoothed MITC3 (ES-MITC3) Method

A novel approach for solving the stiffened shell structures by using an edge-based smoothed MITC3 finite element method (ES-MITC3) is presented in this paper. The ES-MITC3 method is an efficient finite element method by combining the edge-based smoothed finite element method (ES-FEM) with the original MITC3 triangular element to not only significantly improve the accuracy but also overcome the shear-locking phenomenon in the Reissner–Mindlin shell analysis. In this study, the ES-MITC3 method is applied for shell structures and then reinforced by stiffeners based on the Timoshenko beam theory to achieve more durability and strength structures. The transverse displacements of the shell structures and stiffeners at the contact positions are assumed compatible. Numerical results of the ES-MITC3 element are compared with those of available other numerical results to demonstrate a good convergence and accuracy of the present method.

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.