Selective mass scaling for explicit finite element analyses

Due to their inherent lack of convergence problems explicit finite element techniques are widely used for analysing non‐linear mechanical processes. In many such processes the energy content in the high frequency domain is small. By focusing an artificial mass scaling on this domain, the critical time step may be increased substantially without significantly affecting the low frequency behaviour. This is what we refer to as selective mass scaling. Two methods for selective mass scaling are introduced in this work. The proposed methods are based on non‐diagonal mass matrices that scale down the eigenfrequencies of the system. The applicability of the methods is illustrated in two example models where the critical time step is increased by up to 30 times its original size. Copyright © 2005 John Wiley & Sons, Ltd.     

Selective mass scaling for thin walled structures modeled with tri-linear solid elements

A method for selective mass scaling in explicit finite element analyses of thin walled structures, modeled with solid elements, is introduced. The method aims at increasing the critical time step without significantly altering the dynamical response of the system. The proposed method is based on the exclusion of certain rigid body motions from the applied mass scaling by filtering the local velocity field.     

A multiscale mass scaling approach for explicit time integration using proper orthogonal decomposition

One of the main computational issues with explicit dynamics simulations is the significant reduction of the critical time step as the spatial resolution of the finite element mesh increases. In this work, a selective mass scaling approach is presented that can significantly reduce the computational cost in explicit dynamic simulations, while maintaining accuracy. The proposed method is based on a multiscale decomposition approach that separates the dynamics of the system into low (coarse scales) and high frequencies (fine scales). Here, the critical time step is increased by selectively applying mass scaling on the fine scale component only. In problems where the response is dominated by the coarse (low frequency) scales, significant increases in the stable time step can be realized. In this work, we use the proper orthogonal decomposition (POD) method to build the coarse scale space. The main idea behind POD is to obtain an optimal low‐dimensional orthogonal basis for representing an ensemble of high‐dimensional data. In our proposed method, the POD space is generated with snapshots of the solution obtained from early times of the full‐scale simulation. The example problems addressed in this work show significant improvements in computational time, without heavily compromising the accuracy of the results. Copyright © 2013 John Wiley & Sons, Ltd.     

Direct and sparse construction of consistent inverse mass matrices: general variational formulation and application to selective mass scaling

Classical explicit finite element formulations rely on lumped mass matrices. A diagonalized mass matrix enables a trivial computation of the acceleration vector from the force vector. Recently, non‐diagonal mass matrices for explicit finite element analysis (FEA) have received attention due to the selective mass scaling (SMS) technique. SMS allows larger time step sizes without substantial loss of accuracy. However, an expensive solution for accelerations is required at each time step. In the present study, this problem is solved by directly constructing the inverse mass matrix. First, a consistent and sparse inverse mass matrix is built from the modified Hamiltons principle with independent displacement and momentum variables. Usage of biorthogonal bases for momentum allows elimination of momentum unknowns without matrix inversions and directly yields the inverse mass matrix denoted here as reciprocal mass matrix (RMM). Secondly, a variational mass scaling technique is applied to the RMM. It is based on the penalized Hamiltons principle with an additional velocity variable and a free parameter. Using element‐wise bases for velocity and a local elimination yields variationally scaled RMM. Thirdly, examples illustrating the efficiency of the proposed method for simplex elements are presented and discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

A selective mass scaling method for shear wave propagation analyses in nearly incompressible materials

This paper describes a selective mass scaling method which is designed for the analysis of wave propagation problems in nearly incompressible materials. The incompressibility of materials leads to a high value of the compressional wave speed, which makes the time step extremely small in explicit time integration method. The proposed selective mass scaling method selects the eigenfrequencies related to volumetric deformation modes to decrease them, while it keeps the shear eigenmodes unchanged. This makes the time step no longer limited by the compressional wave speed but by the shear wave speed. A significant reduction of CPU time is obtained with a good accuracy for transient problems in small strains on free or largely prestressed media. Copyright © 2016 John Wiley & Sons, Ltd.     

A hybrid stress ANS solid‐shell element and its generalization for smart structure modelling. Part I—solid‐shell element formulation

In the recent years, solid‐shell finite element models which possess no rotational degrees of freedom and applicable to thin plate/shell analyses have attracted considerable attention. Development of these elements are not straightforward. Shear, membrane, trapezoidal, thickness and dilatational lockings must been visioned. In this part of this paper, a novel eight‐node solid‐shell element is proposed. To resolve the shear and trapezoidal lockings, the assumed natural strain (ANS) method is resorted to. The hybrid‐stress formulation is employed to rectify the thickness and dilatational locking. The element is computationally more efficient than the conventional hybrid elements by adopting orthogonal‐assumed stress modes and enforcing admissible sparsity in the flexibility matrix. Popular benchmark tests are exercised to illustrate the efficacy of the elements. In Part II of the paper, the element will be generalized for smart structure modelling by including the piezoelectric effect. Copyright © 2000 John Wiley & Sons, Ltd.     

Efficient and accurate simulation of the packaging forming process

To allow for large‐scale forming applications, such as converting paperboard into package containers, efficient and reliable numerical tools are needed. In finite element simulations of thin structures, elements including structural features are required to reduce the computational cost. Solid‐shell elements based on reduced integration with hourglass stabilization is an attractive choice. One advantage of this choice is the natural inclusion of the thickness, not present in standard degenerated shells, which is especially important for many problems involving contact. Furthermore, no restrictions are imposed on the constitutive models since the solid‐shell element does not require the plane stress condition to be enforced. In this work, a recently proposed efficient solid‐shell element is implemented together with a state‐of‐the‐art continuum model for paperboard. This approach is validated by comparing the obtained numerical results with experimental results for paperboard as well as with those found by using 3D continuum elements. To show the potential of this approach, a large‐scale forming simulation of paperboard is used as a proof of concept.     

Optimal solid shell element for large deformable composite structures with piezoelectric layers and active vibration control

In this paper, we present an optimal low‐order accurate piezoelectric solid‐shell element formulation to model active composite shell structures that can undergo large deformation and large overall motion. This element has only displacement and electric degrees of freedom (dofs), with no rotational dofs, and an optimal number of enhancing assumed strain (EAS) parameters to pass the patch tests (both membrane and out‐of‐plane bending). The combination of the present optimal piezoelectric solid‐shell element and the optimal solid‐shell element previously developed allows for efficient and accurate analyses of large deformable composite multilayer shell structures with piezoelectric layers. To make the 3‐D analysis of active composite shells containing discrete piezoelectric sensors and actuators even more efficient, the composite solid‐shell element is further developed here. Based on the mixed Fraeijs de Veubeke–Hu–Washizu (FHW) variational principle, the in‐plane and out‐of‐plane bending behaviours are improved via a new and efficient enhancement of the strain tensor. Shear‐locking and curvature thickness locking are resolved effectively by using the assumed natural strain (ANS) method. We also present an optimal‐control design for vibration suppression of a large deformable structure based on the general finite element approach. The linear‐quadratic regulator control scheme with output feedback is used as a control law on the basis of the state space model of the system. Numerical examples involving static analyses and dynamic analyses of active shell structures having a large range of element aspect ratios are presented. Active vibration control of a composite multilayer shell with distributed piezoelectric sensors and actuators is performed to test the present element and the control design procedure. Copyright © 2005 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.     

A simplified co‐rotational method for quadrilateral shell elements in geometrically nonlinear analysis

This paper presents a simplified co‐rotational formulation for quadrilateral shell elements inheriting the merit of element‐independence from the traditional co‐rotational approach in literature. With the objective of application to nonlinear analysis of civil engineering structures, the authors further simplify the formulation of the geometrical stiffness using the small strain assumption, which is valid in the co‐rotational approach, with the warping effects considered as eccentricities. Compared with the traditional element‐independent co‐rotational method, the projector is neglected both in the tangent stiffness matrix and in the internal force vector for simplicity in formulation. Meanwhile, a quadrilateral flat shell element allowing for drilling rotations is adopted and incorporated into this simplified co‐rotational algorithm for geometrically nonlinear analysis involved with large displacements and large rotations. Several benchmark problems are presented to confirm the efficiency and accuracy of the proposed method for practical applications.     

An enhanced co‐rotational approach for large displacement analysis of plates

This paper presents a new co‐rotational approach for the large displacement analysis of plates employing 4‐noded quadrilateral flat shell elements. The proposed approach benefits from (i) a simple local co‐rotational system invariant to the element nodal ordering, (ii) the choice of the two smallest components of the nodal normal vector as global rotational degrees of freedom, and (iii) the use of hierarchic freedoms, that are unaffected by the co‐rotational transformations, for higher‐order accuracy. Important additional benefits that arise from the aforementioned features include symmetry of the tangent stiffness matrix and complete insensitivity of the large displacement transformations to the size of the incremental step. The applicability of the new approach to moderately thick as well as thin plates is illustrated by considering two alternative local formulations based on the Reissner–Mindlin and discrete Kirchhoff hypotheses. Several examples are finally presented which demonstrate the accuracy, step‐insensitivity and computational benefits of the proposed co‐rotational approach for large displacement analysis of plate structures. Copyright © 2005 John Wiley & Sons, Ltd.     

A new one‐point quadrature enhanced assumed strain (EAS) solid‐shell element with multiple integration points along thickness: Part I—geometrically linear applications

Accuracy and efficiency are the main features expected in finite element method. In the field of low‐order formulations, the treatment of locking phenomena is crucial to prevent poor results. For three‐dimensional analysis, the development of efficient and accurate eight‐node solid‐shell finite elements has been the principal goal of a number of recent published works. When modelling thin‐ and thick‐walled applications, the well‐known transverse shear and volumetric locking phenomena should be conveniently circumvented. In this work, the enhanced assumed strain method and a reduced in‐plane integration scheme are combined to produce a new eight‐node solid‐shell element, accommodating the use of any number of integration points along thickness direction. Furthermore, a physical stabilization procedure is employed in order to correct the element's rank deficiency. Several factors contribute to the high computational efficiency of the formulation, namely: (i) the use of only one internal variable per element for the enhanced part of the strain field; (ii) the reduced integration scheme; (iii) the prevention of using multiple elements' layers along thickness, which can be simply replaced by any number of integration points within a single element layer. Implementation guidelines and numerical results confirm the robustness and efficiency of the proposed approach when compared to conventional elements well‐established in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

A non‐linear interface element for 3D mesoscale analysis of brick‐masonry structures

This paper presents a novel interface element for the geometric and material non‐linear analysis of unreinforced brick‐masonry structures. In the proposed modelling approach, the blocks are modelled using 3D continuum solid elements, whereas the mortar and brick–mortar interfaces are modelled by means of the 2D non‐linear interface element. This enables the representation of any 3D arrangement for brick‐masonry, accounting for the in‐plane stacking mode and the through‐thickness geometry, and importantly it allows the investigation of both the in‐plane and the out‐of‐plane responses of unreinforced masonry panels. A co‐rotational approach is employed for the interface element, which shifts the treatment of geometric non‐linearity to the level of discrete entities, and enables the consideration of material non‐linearity within a simplified local framework employing first‐order kinematics. In this respect, the internal interface forces are modelled by means of elasto‐plastic material laws based on work‐softening plasticity and employing multi‐surface plasticity concepts. Following the presentation of the interface element formulation details, several experimental–numerical comparisons are provided for the in‐plane and out‐of‐plane static behaviours of brick‐masonry panels. The favourable results achieved demonstrate the accuracy and the significant potential of using the developed interface element for the non‐linear analysis of brick‐masonry structures under extreme loading conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

On immunizing five‐beta hybrid‐stress element models from ‘trapezoidal locking’ in practical analyses

When multiple trapezoidal four‐node plane elements are used to model slender beams, it is found that the transverse bending stress/strain mode which should physically vanish is most detrimental to the element accuracy and leads to a deficiency which is sometimes known as ‘trapezoidal locking’. In this paper, immunity of four‐node five‐beta hybrid elements to ‘trapezoidal locking’ in practical analyses is obtained by a simple selective scaling procedure that judiciously reduces the stiffness arising from the two bending stress/strain modes in the elements. Copyright © 2000 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.     

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.     

On consistent discretization of inertia terms for C0 structural elements with modified stiffness matrices

In the existing finite element calculations of dynamic problems using C⁰ structural elements, the inertia terms are evaluated without any reference to the modifications such as reduced integration, projections etc., typically needed in the discretization of the stiffness terms. A different discretization of inertia is discussed here. It is based on the following two observations. First, as shown in this work (at least for the beam problems), the modified stiffness matrix for a given C⁰ element can be obtained by standard, unmodified approach, in which degrees of freedom remain unchanged, but the shape functions are different. Those modified functions are of higher order and define the translational field within the element in terms of both translational and rotational parameters. Second, if standard consistent approach to the formulation of dynamic problems is to be followed, approximation of the displacement field used in the unmodified evaluation of the stiffness terms should also be used in discretization of the inertia terms. This implies that the modified higher-order functions should be employed when evaluating the element mass matrix for the C⁰ elements with modified stiffness matrices. As a consequence of this approach, consistency between formulation of the inertia and stiffness terms is restored. This leads to inertial coupling between rotational and translational degrees of freedom, which is absent in standard evaluation of inertia. It is demonstrated that this approach tends to improve accuracy of dynamic computations. © 1997 by John Wiley & Sons, Ltd.     

Development of shear locking‐free shell elements using an enhanced assumed strain formulation

The degenerated approach for shell elements of Ahmad and co‐workers is revisited in this paper. To avoid transverse shear locking effects in four‐node bilinear elements, an alternative formulation based on the enhanced assumed strain (EAS) method of Simo and Rifai is proposed directed towards the transverse shear terms of the strain field. In the first part of the work the analysis of the null transverse shear strain subspace for the degenerated element and also for the selective reduced integration (SRI) and assumed natural strain (ANS) formulations is carried out. Locking effects are then justified by the inability of the null transverse shear strain subspace, implicitly defined by a given finite element, to properly reproduce the required displacement patterns. Illustrating the proposed approach, a remarkably simple single‐element test is described where ANS formulation fails to converge to the correct results, being characterized by the same performance as the degenerated shell element. The adequate enhancement of the null transverse shear strain subspace is provided by the EAS method, enforcing Kirchhoff hypothesis for low thickness values and leading to a framework for the development of shear‐locking‐free shell elements. Numerical linear elastic tests show improved results obtained with the proposed formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

层合板层间断裂分析的组合界面单元及其特性

提出了一种由刚性元和零厚度的内聚力单元组合而成的新型界面单元,该界面单元嵌在板壳结构界面之间,可用来模拟界面损伤的起始和演化,能考虑板壳的平动和转动对分层损伤的作用。该界面单元具有有限厚度,八个结点,每个结点有五个自由度,通过刚性元将板壳单元结点的位移和结点力转换到内部零厚度的内聚力单元上,界面损伤通过内聚力单元的损伤演化体现出来。采用板壳单元和新型界面单元建立有限元模型,对混合弯曲(MMB)试验和双悬臂梁(DCB)弯曲试验进行了计算模拟,计算结果能很好地模拟结构的界面损伤过程。相比传统的用内聚力单元和三维实体单元组成的模型,建模方便,在精度相当的前提下,可以使单元尺寸增大一倍,减少裂尖内聚力区域(cohesive zone)内的单元数量,缩小计算规模,提高计算效率。