Modal validation of sandwich shell finite elements based on a p‐order shear deformation theory including zigzag terms

This paper describes a set of improved C⁰‐compatible composite shell finite elements for evaluating the global dynamic response (natural frequencies and mode shapes) of sandwich structures. Combining a through‐the‐thickness displacement approximation of variable high order with a first‐order zigzag function, the proposed finite elements are suited for modelling sandwich plates and doubly curved shells with a non‐uniform thickness and are more accurate than conventional models based on the first‐ and third‐order shear deformation theories, especially in sandwich panels with highly heterogeneous properties. The new finite element model is then validated by a comparison with the standard shell and 3D solid models. From these investigations, it can be concluded that adding a zigzag function even to high‐order polynomial approximations of the through‐the‐thickness displacement is a useful tool for refining the modelling of sandwich structures. In addition, the proposed formulation is sufficiently versatile to represent with the same level of accuracy the behaviour of thin‐to‐thick laminated shells as well as of strongly heterogeneous sandwich structures. Copyright © 2008 John Wiley & Sons, Ltd.     

3D‐based hp‐adaptive first‐order shell finite element for modelling and analysis of complex structures—Part 2: Application to structural analysis

The paper presents a 3D‐based adaptive first‐order shell finite element to be applied to hierarchical modelling and adaptive analysis of complex structures. The main feature of the element is that it is equipped with 3D degrees of freedom, while its mechanical model corresponds to classical first‐order shell theory. Other useful features of the element are its modelling and adaptive capabilities. The element is assigned to hierarchical modelling and hpq‐adaptive analysis of shell parts of complex structures consisting of solid, thick‐ and thin‐shell parts, as well as of transition zones, where h, p and q denote the mesh density parameter and the longitudinal and transverse orders of approximation, respectively. The proposed hp‐adaptive first‐order shell element can be joined with 3D‐based hpq‐adaptive hierarchical shell elements or 3D hpp‐adaptive solid elements by means of the family of 3D‐based hpq/hp‐ or hpp/hp‐adaptive transition elements. The main objective of the first part of our research, presented in the first part of the paper, was to provide non‐standard information on the original parts of the element algorithm. Here we describe the second part of the research, devoted to the methodology and results of the application of the element to various plate and shell problems. The main objective of this part is to verify algorithms of the element and to show its usefulness in modelling and adaptive analysis of shell and plate parts of complex structures. In order to do that, there is a presentation of the results of a comparative analysis of model plate and shell problems using the classical and our elements, and equidistributed and integrated Legendre shape functions. For the plate problem a comparison of the results obtained from the adaptive and non‐adaptive analysis is also included. Additionally, some advantages of the application of our element are shown through a comparative analysis of p‐convergence of the thin plate problem and an adaptive analysis of the exemplary complex structure. Copyright © 2006 John Wiley & Sons, Ltd.     

A new one‐point quadrature enhanced assumed strain (EAS) solid‐shell element with multiple integration points along thickness—part II: nonlinear applications

In this work the recently proposed Reduced Enhanced Solid‐Shell (RESS) finite element, based on the enhanced assumed strain (EAS) method and a one‐point quadrature integration scheme, is extended in order to account for large deformation elastoplastic thin‐shell problems. One of the main features of this finite element consists in its minimal number of enhancing parameters (one), sufficient to circumvent the well‐known Poisson and volumetric locking phenomena, leading to a computationally efficient performance when compared to other 3D or solid‐shell enhanced strain elements. Furthermore, the employed numerical integration accounts for an arbitrary number of integration points through the thickness direction within a single layer of elements. The EAS formulation comprises an additive split of the Green–Lagrange material strain tensor, making the inclusion of nonlinear kinematics a straightforward task. A corotational coordinate system is used to integrate the constitutive law and to ensure incremental objectivity. A physical stabilization procedure is implemented in order to correct the element's rank deficiencies. A variety of shell‐type numerical benchmarks including plasticity, large deformations and contact are carried out, and good results are obtained when compared to well‐established formulations in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

3D‐based hp‐adaptive first‐order shell finite element for modelling and analysis of complex structures—Part 1: The model and the approximation

The paper presents a 3D‐based adaptive first‐order shell finite element to be applied to hierarchical modelling and adaptive analysis of complex structures. The main feature of the element is that it is equipped with 3D degrees of freedom, while its mechanical model corresponds to classical first‐order shell theory. Other useful features of the element are its modelling and adaptive capabilities. The element is assigned to hierarchical modelling and hpq‐adaptive analysis of shell parts of complex structures consisting of solid, thick‐ and thin‐shell parts, as well as of transition zones, where h, p and q denote the mesh density parameter and the longitudinal and transverse orders of approximation, respectively. The proposed hp‐adaptive first‐order shell element can be joined with 3D‐based hpq‐adaptive hierarchical shell elements or 3D hpp‐adaptive solid elements by means of the family of 3D‐based hpq/hp‐ or hpp/hp‐adaptive transition elements. The main objective of the first part of our research, presented in this paper, is to provide non‐standard information on the original parts of the element algorithm. In order to do that, we present the definition of shape functions necessary for p‐adaptivity, as well as the procedure for imposing constraints corresponding to the lack of elongation of the straight lines perpendicular to the shell mid‐surface, which is the procedure necessary for q‐adaptivity. The 3D version of constrained approximation presented next is the basis for h‐adaptivity of the element. The second part of our research, devoted to methodology and results of the numerical research on application of the element to various plate and shell problems, are described in the second part of this paper. Copyright © 2006 John Wiley & Sons, Ltd.     

X‐FEM in isogeometric analysis for linear fracture mechanics

The extended finite element method (X‐FEM) has proven to be an accurate, robust method for solving problems in fracture mechanics. X‐FEM has typically been used with elements using linear basis functions, although some work has been performed using quadratics. In the current work, the X‐FEM formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics. In comparison with X‐FEM with conventional finite elements of equal degree, the NURBS‐based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom (DOF). Results for linear through quartic NURBS basis functions are presented for a multiplicity of one or a multiplicity equal the degree. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient and accurate multilayer solid‐shell element: non‐linear materials at finite strain

We present in this paper an efficient and accurate low‐order solid‐shell element formulation for analyses of large deformable multilayer shell structures with non‐linear materials. The element has only displacement degrees of freedom (dofs), and an optimal number of enhancing assumed strain (EAS) parameters to pass the patch tests (both membrane and out‐of‐plane bending) and to remedy volumetric locking. Based on the mixed Fraeijs de Veubeke‐Hu‐Washizu (FHW) variational principle, the in‐plane and out‐of‐plane bending behaviours are improved and the locking associated with (nearly) incompressible materials is avoided via a new efficient enhancement of strain tensor. Shear locking and curvature thickness locking are resolved effectively by using the assumed natural strain (ANS) method. Two non‐linear 3‐D constitutive models (Mooney–Rivlin material and hyperelastoplastic material at finite strain) are applied directly without requiring the enforcement of the plane‐stress assumption. In particular, we give a simple derivation for the hyperelastoplastic model using spectral representations. In addition, the present element has a well‐defined lumped mass matrix, and provides double‐side contact surfaces for shell contact problems. With the dynamics referred to a fixed inertial frame, the present element can be used to analyse multilayer shell structures undergoing large overall motion. Numerical examples involving static analyses and implicit/explicit dynamic analyses of multilayer shell structures with both material and geometric non‐linearities are presented, and compared with existing results obtained from other shell elements and from a meshless method. It is shown that elements that did not pass the out‐of‐plane bending patch test could not provide accurate results, as compared to the present element formulation, which passed the out‐of‐plane bending patch test. The present element proves to be versatile and efficient in the modelling and analyses of general non‐linear composite multilayer shell structures. Copyright © 2005 John Wiley & Sons, Ltd.     

Analysis of a rotation‐free 4‐node shell element

In this paper, a shell element for small and large deformations is presented based on the extension of the methodology to derive triangular shell element without rotational degrees of freedom (so‐called rotation‐free). As in our original triangular S3 element, the curvatures are computed resorting to the surrounding elements. However, the extension to a quadrilateral element requires internal curvatures in order to avoid singular bending stiffness. The quadrilateral area co‐ordinates interpolation is used to establish the required expressions between the rigid‐body modes of normal nodal translations and the normal through thickness bending strains at mid‐side. In order to propose an attractive low‐cost shell element, the one‐point quadrature is achieved at the centre for the membrane strains, which are superposed to the bending strains in the centred co‐rotational local frame. The membrane hourglass control is obtained by the perturbation stabilization procedure. Free, simply supported and clamped edges are considered without introducing virtual nodes or elements. Several numerical examples with regular and irregular meshes are performed to show the convergence, accuracy and the reasonable little sensitivity to geometric distortion. Based on an updated Lagrangian formulation and Newton iterations, the large displacements of the pinched hemispherical shell show the effectiveness of the proposed simplified element (S4). Finally, the deep drawing of a square box including large plastic strains with contact and friction completes the ability of the rotation‐free quadrilateral element for sheet‐metal‐forming simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

A large deformation frictional contact formulation using NURBS‐based isogeometric analysis

This paper focuses on the application of NURBS‐based isogeometric analysis to Coulomb frictional contact problems between deformable bodies, in the context of large deformations. A mortar‐based approach is presented to treat the contact constraints, whereby the discretization of the continuum is performed with arbitrary order NURBS, as well as C⁰‐continuous Lagrange polynomial elements for comparison purposes. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. Results of lower quality are obtained from the Lagrange discretization, as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive through‐thickness integration for accurate springback prediction

Accurate numerical prediction of springback in sheet metal forming is essential for the automotive industry. Numerous factors influence the accuracy of prediction of this complex phenomenon by using the finite element method. One of them is the numerical integration through the thickness of shell elements. It is known that the traditional numerical schemes are very inefficient in elastic–plastic analysis and even for simple problems they require up to 50 integration points for an accurate springback prediction. An adaptive through‐thickness integration strategy can be a good alternative. The main characteristic feature of the strategy is that it defines abscissas and weights depending on the integrand's properties and, thus, can adapt itself to improve the accuracy of integration. A concept of an adaptive through‐thickness integration strategy for shell elements is presented in this paper. Its potential is demonstrated using two examples. Calculations of a simple test—bending a beam under tension—show that for a similar set of material and process parameters the adaptive rule with seven integration points performs significantly better than the traditional trapezoidal rule with 50 points. Simulations of an unconstrained cylindrical bending problem demonstrate that the adaptive through‐thickness integration strategy for shell elements can guarantee an accurate springback prediction at minimal costs. Copyright © 2007 John Wiley & Sons, Ltd.     

An eight‐node hybrid‐stress solid‐shell element for geometric non‐linear analysis of elastic shells

This paper presents eight‐node solid‐shell elements for geometric non‐linear analysis of elastic shells. To subdue shear, trapezoidal and thickness locking, the assumed natural strain method and an ad hoc modified generalized laminate stiffness matrix are employed. A selectively reduced integrated element is formulated with its membrane and bending shear strain components taken to be constant and equal to the ones evaluated at the element centroid. With the generalized stresses arising from the modified generalized laminate stiffness matrix assumed to be independent from the ones obtained from the displacement, an extended Hellinger–Reissner functional can be derived. By choosing the assumed generalized stresses similar to the assumed stresses of a previous solid element, a hybrid‐stress solid‐shell element is formulated. Commonly employed geometric non‐linear homogeneous and laminated shell problems are attempted and our results are close to those of other state‐of‐the‐art elements. Moreover, the hybrid‐stress element converges more readily than the selectively reduced integrated element in all benchmark problems. Copyright © 2002 John Wiley & Sons, Ltd.     

An improved solid‐shell element based on ANS and EAS concepts

This paper presents an eight‐node nonlinear solid‐shell element for static problems. The main goal of this work is to develop a solid‐shell formulation with improved membrane response compared with the previous solid‐shell element (MOS2013), presented in 1 . Assumed natural strain concept is implemented to account for the transverse shear and thickness strains to circumvent the curvature thickness and transverse shear locking problems. The enhanced assumed strain approach based on the Hu–Washizu variational principle with six enhanced assumed strain degrees of freedom is applied. Five extra degrees of freedom are applied on the in‐plane strains to improve the membrane response and one on the thickness strain to alleviate the volumetric and Poisson's thickness locking problems. The ensuing element performs well in both in‐plane and out‐of‐plane responses, besides the simplicity of implementation. The element formulation yields exact solutions for both the membrane and bending patch tests. The formulation is extended to the geometrically nonlinear regime using the corotational approach, explained in 2 . Numerical results from benchmarks show the robustness of the formulation in geometrically linear and nonlinear problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Elasto‐plastic finite element analysis of shells with damage due to microvoids

This paper presents a non‐linear finite element analysis for the elasto‐plastic behaviour of thick/thin shells and plates with large rotations and damage effects. The refined shell theory given by Voyiadjis and Woelke (Int. J. Solids Struct. 2004; 41 :3747–3769) provides a set of shell constitutive equations. Numerical implementation of the shell theory leading to the development of the C⁰ quadrilateral shell element (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted) is used here as an effective tool for a linear elastic analysis of shells. The large rotation elasto‐plastic model for shells presented by Voyiadjis and Woelke (General non‐linear finite element analysis of thick plates and shells. 2006, submitted) is enhanced here to account for the damage effects due to microvoids, formulated within the framework of a micromechanical damage model. The evolution equation of the scalar porosity parameter as given by Duszek‐Perzyna and Perzyna (Material Instabilities: Theory and Applications, ASME Congress, Chicago, AMD‐Vol. 183/MD‐50, 9–11 November 1994; 59–85) is reduced here to describe the most relevant damage effects for isotropic plates and shells, i.e. the growth of voids as a function of the plastic flow. The anisotropic damage effects, the influence of the microcracks and elastic damage are not considered in this paper. The damage modelled through the evolution of porosity is incorporated directly into the yield function, giving a generalized and convenient loading surface expressed in terms of stress resultants and stress couples. A plastic node method (Comput. Methods Appl. Mech. Eng. 1982; 34 :1089–1104) is used to derive the large rotation, elasto‐plastic‐damage tangent stiffness matrix. Some of the important features of this paper are that the elastic stiffness matrix is derived explicitly, with all the integrals calculated analytically (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted). In addition, a non‐layered model is adopted in which integration through the thickness is not necessary. Consequently, the elasto‐plastic‐damage stiffness matrix is also given explicitly and numerical integration is not performed. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power and time. Copyright © 2006 John Wiley & Sons, Ltd.     

Adaptive modelling of delamination initiation and propagation using an equivalent single‐layer shell approach

A major challenge for crash failure analysis of laminated composites is to find a modelling approach, which is both sufficiently accurate, for example, able to capture delaminations, and computationally efficient to allow full‐scale vehicle crash simulations. Addressing this challenge, we propose a methodology based on an equivalent single‐layer shell formulation which is adaptively through‐the‐thickness refined to capture initiating and propagating delaminations. To be specific, single shell elements through the laminate thickness are locally and adaptively enriched using the extended finite element method such that delaminations can be explicitly modelled without having to be represented by separate elements. Furthermore, the shell formulation is combined with a stress recovery technique which increases the accuracy of predicting delamination initiations. The paper focuses on the parameters associated with identifying, introducing and extending the enrichment areas; especially on the impact of these parameters on the resulting structural deformation behaviour. We conclude that the delamination enrichment must be large enough to allow the fracture process to be accurately resolved, and we propose a suitable approach to achieve this. The proposed methodology for adaptive delamination modelling shows potential for being computationally efficient, and thereby, it has the potential to enable efficient and accurate full vehicle crash simulations of laminated composites. Copyright © 2017 John Wiley & Sons, Ltd.     

NURBS based least‐squares finite element methods for fluid and solid mechanics

This contribution investigates the performance of a least‐squares finite element method based on non‐uniform rational B‐splines (NURBS) basis functions. The least‐squares functional is formulated directly in terms of the strong form of the governing equations and boundary conditions. Thus, the introduction of auxiliary variables is avoided, but the order of the basis functions must be higher or equal to the order of the highest spatial derivatives. The methodology is applied to the incompressible Navier–Stokes equations and to linear as well as nonlinear elastic solid mechanics. The numerical examples presented feature convective effects and incompressible or nearly incompressible material. The numerical results, which are obtained with equal‐order interpolation and without any stabilisation techniques, are smooth and accurate. It is shown that for p and h refinement, the theoretical rates of convergence are achieved. Copyright © 2014 John Wiley & Sons, Ltd.     

Selective mass scaling for distorted solid‐shell elements in explicit dynamics: optimal scaling factor and stable time step estimate

The use of solid‐shell elements in explicit dynamics has been so far limited by the small critical time step resulting from the small thickness of these elements in comparison with the in‐plane dimensions. To reduce the element highest eigenfrequency in inertia dominated problems, the selective mass scaling approach previously proposed in [G. Cocchetti, M. Pagani and U. Perego, Comp. & Struct. 2013; 127:39‐52.] for parallelepiped elements is here reformulated for distorted solid‐shell elements. The two following objectives are achieved: the critical time step is governed by the smallest element in‐plane dimension and not anymore by the thickness; the mass matrix remains diagonal after the selective mass scaling. The proposed approach makes reference to one Gauss point, trilinear brick element, for which the maximum eigenfrequency can be computed analytically. For this element, it is shown that the proposed mass scaling can be interpreted as a geometric thickness scaling, obtaining in this way a simple criterion for the definition of the optimal mass scaling factor. A strategy for the effective computation of the element maximum eigenfrequency is also proposed. The considered mass scaling preserves the element translational inertia, while it modifies the rotational one, leading to errors in the kinetic energy when the motion rotational component is dominant. The error has been rigorously assessed for an individual element, and a simple formula for its estimate has been derived. Numerical tests, both in small and large displacements and rotations, using a state‐of‐the‐art solid‐shell element taken from the literature, confirm the effectiveness and accuracy of the proposed approach. Copyright © 2014 John Wiley & Sons, Ltd.     

3D NURBS‐enhanced finite element method (NEFEM)

This paper presents the extension of the recently proposed NURBS‐enhanced finite element method (NEFEM) to 3D domains. NEFEM is able to exactly represent the geometry of the computational domain by means of its CAD boundary representation with non‐uniform rational B‐splines (NURBS) surfaces. Specific strategies for interpolation and numerical integration are presented for those elements affected by the NURBS boundary representation. For elements not intersecting the boundary, a standard finite element rationale is used, preserving the efficiency of the classical FEM. In 3D NEFEM special attention must be paid to geometric issues that are easily treated in the 2D implementation. Several numerical examples show the performance and benefits of NEFEM compared with standard isoparametric or cartesian finite elements. NEFEM is a powerful strategy to efficiently treat curved boundaries and it avoids excessive mesh refinement to capture small geometric features. Copyright © 2011 John Wiley & Sons, Ltd.