Assumed Strain Method Research Articles

Mean-strain 8-node hexahedron with optimized energy-sampling stabilization

A method for stabilizing the mean-strain hexahedron was described by Krysl (in IJNME 2014). The technique relied on a sampling of the stabilization energy using the mean-strain quadrature and the full Gaussian integration rule, which was shown to guarantee consistency and stability. The stabilization energy was assumed to be generated by a modified constitutive matrix based on the spectral decomposition. The stabilization required user-selected values of the stabilization parameters. In the present work we eliminate the arbitrariness of the stabilization parameters. We formulate the technique more precisely as an assumed-strain method, and we express the stabilization energy in terms of input parameters of the real material. Finally, we fix the value of the stabilization parameters in a quasi-optimal manner by linking the stabilization to the bending behavior of the hexahedral element. For simplicity the developments are limited to linear elasticity, but with an arbitrarily anisotropic elasticity matrix. The accuracy and convergence characteristics of the present formulations compare favorably with the capabilities of mean-strain and other high-performance hexahedral elements as implemented in Abaqus and with a number of successful hexahedral and shell elements and we demonstrate that the present element performs very well when used with large aspect ratios for thin structures such as plates or shells.

A new reduced integration solid‐shell element based on EAS and ANS with hourglass stabilization

SummaryIn this paper, a novel reduced integration eight‐node solid‐shell finite element formulation with hourglass stabilization is proposed. The enhanced assumed strain method is adopted to eliminate the well‐known volumetric and Poisson thickness locking phenomena with only one internal variable required. In order to alleviate the transverse shear and trapezoidal locking and correct rank deficiency simultaneously, the assumed natural strain method is implemented in conjunction with the Taylor expansion of the inverse Jacobian matrix. The projection of the hourglass strain‐displacement matrix and reconstruction of its transverse shear components are further employed to avoid excessive hourglass stiffness. The proposed solid‐shell element formulation successfully passes both the membrane and bending patch tests. Several typical examples are presented to demonstrate the excellent performance and extensive applicability of the proposed element. Copyright © 2015 John Wiley & Sons, Ltd.

Finite rotation FE-simulation and active vibration control of smart composite laminated structures

The present article focuses on the nonlinear finite element simulation and control of large amplitude vibrations of smart piezolaminated composite structures. Full geometrically nonlinear finite rotation strain---displacement relations and Reissner---Mindlin first-order shear deformation hypothesis to include the transverse shear effects are considered to derive the variational formulation. A quadratic variation of electric potential is assumed in transverse direction. An assumed natural strain method for the shear strains, an enhanced assumed strain method for the membrane strains and an enhanced assumed gradient method for the electric field is incorporated to improve the behavior of a four-node shell element. Numerical simulations presented in this article show the accurate prediction capabilities of the proposed method, especially for structures undergoing finite deformations and rotations, in comparison to the results obtained by simplified nonlinear models available in references and also with those obtained by using the C3D20RE solid element for piezoelectric layers in the Abaqus code.

On the use of elastic middle surface approach in the large deformation analysis of moderately thick shell structures using absolute nodal coordinate formulation

In this study, a shear deformable shell element is developed based on the elastic middle surface approach using the absolute nodal coordinate formulation (ANCF) for the large deformation analysis of thin to moderately thick shell structures. The bilinear shape function is used to define the global position vector in the middle surface and the transverse gradient vector which defines the orientation and deformation of the cross section within the element. The plane stress assumption is used to remedy the Poisson’s thickness locking exhibited in the ANCF shell element formulated by the continuum mechanics approach, thus the stress distribution along the shell thickness is assumed to be constant. The cross-sectional frame is introduced to define strains of the initially curved shell element using the elastic middle surface approach. The curvature thickness and transverse shear lockings are alleviated using the assumed natural strain method, while the in-plane shear locking is removed using the enhanced assumed strain method. Several numerical examples are presented in order to demonstrate the performance of the shear deformable ANCF shell element based on the elastic middle surface approach developed in this study. The developed element is compared with the continuum mechanics-based ANCF shell element to shed light on the nature of the thickness locking exhibited in the bilinear shell element and its locking remedies.

A thermodynamically consistent and numerically stable formulation for the description of diffusion in polymeric gels

AbstractThis work proposes a stable equal‐order element formulation for polymeric gels at finite deformations. From the numerical point of view the description of flow in incompressible elastic media such as gels leads to singularities in the coupled system of equations of the corresponding finite element discretization unless certain requirements are met to satisfy the Ladyženskaja‐Babuška‐Brezzi (LBB) condition. The proposed computationally efficient mixed enhanced assumed strain method allows to satisfy the LBB condition and to account for locking phenomena without the requirement of higher order interpolations for the displacement field. The proposed model can be employed in various fields where the modeling of flow in incompressible media undergoing finite deformations is of interest such as Computational Soil Mechanics or Computational Biomechanics. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A solid-beam finite element and non-linear constitutive modelling

The successful concept of solid-shell formulations which brings up several advantages with respect to classical shell elements is transferred to beam structures. The resulting three-dimensional solid-beam element possesses eight nodes with only displacement degrees-of-freedom. A reduced integration concept is worked out which accounts for the presence of two thickness directions. The relevant locking phenomena are treated by a combination of the assumed natural strain method and the enhanced assumed strain method. It is shown at which points the steps to derive the solid-beam formulation deviate from the derivation of the corresponding solid-shell. Validations for elasticity and elasto-plasticity with a reference to classical beam finite elements are provided. In particular beam structures with arbitrary four-sided cross-section can efficiently be modelled by using only one element within the cross-section. If more complex cross-sections are modelled by using several elements within the cross-section, the formulation shows a superior performance compared with standard solid elements.

An enhanced cell‐based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh

SUMMARYIn this work, an enhanced cell‐based smoothed finite element method (FEM) is presented for the Reissner–Mindlin plate bending analysis. The smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed FEM is reformulated, and the relationship between the original approach and the present method in curvature smoothing is established. To improve the accuracy of shear strain in a distorted mesh, we span the shear strain space over the adjacent element. This is performed by employing an edge‐based smoothing technique through a simple area‐weighted smoothing procedure on MITC4 assumed shear strain field. A three‐field variational principle is utilized to develop the mixed formulation. The resultant element formulation is further reduced to a displacement‐based formulation via an assumed strain method defined by the edge‐smoothing technique. As the result, a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement/rotation fields and smoothing operators can be shown to improve the solution accuracy in cell‐based smoothed FEM for Reissner–Mindlin plate bending analysis. Several numerical examples are presented to demonstrate the accuracy of the proposed formulation.Copyright © 2013 John Wiley & Sons, Ltd.

On the modeling and design of composite multilayered structures using solid-shell finite element model

In this investigation a coupling between a 3D solid-shell element for the analysis of multilayered composite shell structures and a specific response surface method is proposed. The first part of the paper is dedicated to the finite element formulation of a developed composite 8-node solid-shell element called SCH8γ7, based only on translational degrees of freedom. The basis of the present finite element formulation is the standard 8-node brick element with tri-linear shape functions. A particular attention is given to alleviate shear, trapezoidal and thickness locking, without resorting to the classical plane-stress assumption. Assumed natural strain method and enhanced assumed strain method are used to improve the relative poor element behavior of a standard hexahedral displacement element. The anisotropic material behavior of layered shells is modeled using a fully three dimensional elastic orthotropic material law in each layer, including the thickness stress component. The second part of the paper will focus on an adaptive response surface method for the structural optimization problem. The response surfaces are built using moving least squares approximations and design of experiments by means of a specific method called Diffuse Approximation.Several numerical applications to composite multilayered shell structures are studied to show the applicability and effectiveness of the proposed procedure. Good results of analysis and optimization using the developed SCH8γ7 solid-shell element have been obtained in comparison with reference analytical solutions and with those obtained using the SC8R solid-shell finite element available in ABAQUS© code.

A four‐node corotational quadrilateral elastoplastic shell element using vectorial rotational variables

SUMMARYA four‐node corotational quadrilateral elastoplastic shell element is presented. The local coordinate system of the element is defined by the two bisectors of the diagonal vectors generated from the four corner nodes and their cross product. This local coordinate system rotates rigidly with the element but does not deform with the element. As a result, the element rigid‐body rotations are excluded in calculating the local nodal variables from the global nodal variables. The two smallest components of each nodal orientation vector are defined as rotational variables, leading to the desired additive property for all nodal variables in a nonlinear incremental solution procedure. Different from other existing corotational finite‐element formulations, the resulting element tangent stiffness matrix is symmetric owing to the commutativity of the local nodal variables in calculating the second derivative of strains with respect to these variables. For elastoplastic analyses, the Maxwell–Huber–Hencky–von Mises yield criterion is employed, together with the backward‐Euler return‐mapping method, for the evaluation of the elastoplastic stress state; the consistent tangent modulus matrix is derived. To eliminate locking problems, we use the assumed strain method. Several elastic patch tests and elastoplastic plate/shell problems undergoing large deformation are solved to demonstrate the computational efficiency and accuracy of the proposed formulation. Copyright © 2013 John Wiley & Sons, Ltd.

Two improvements in formulation of nine‐node element MITC9

SUMMARYThe paper concerns a well‐known two‐dimensional nine‐node quadrilateral element MITC9, which is based on two‐level approximations of strains (assumed strain method). The element has good accuracy, but does not pass the patch test.As the first improvement, we propose a modification of the element's transformations, partly resolving the problem with the patch test. The source of the problem is the use of covariant components in a (local) natural co‐basis, different at each sampling point.As the second improvement, we use the corrected shape functions of Celia MA, Gray WG. An improved isoparametric transformation for finite element analysis. International Journal for Numerical Methods in Engineering 1984; 20:1447–1459, extending their applicability to the nine‐node element for plane elasticity and the 3 × 3 integration. Originally, they are tested for an eight‐node element for the heat conduction equation and the 4 × 4 integration.The improved element, designated as MITC9i, is based on the Green strain and derived from the potential energy for the plane stress condition. It is subjected to a range of tests, to confirm that it passes the patch test for several types of mesh distortions, to prove its coarse mesh accuracy and the absence of locking as well as to establish its sensitivity to mesh distortions.The improved element MITC9i performs substantially better than the MITC9 element, QUAD9** element, and our previous 9‐AS element.Copyright © 2012 John Wiley & Sons, Ltd.

A new prismatic solid-shell element 'SHB6' : assumed-strain formulation and evaluation on benchmark problems

In this paper, the formulation of a new six-node solid–shell element denoted (SHB6) is proposed. This prismatic element is based on a purely three-dimensional approach, and hence has displacements as the only degrees of freedom. A reduced integration scheme is adopted consisting of one-point in-plane quadrature and an arbitrary number of integration points, with a minimum number of two, distributed along the ‘thickness’ direction. Moreover, in order to enhance its performance and to greatly reduce most locking effects, specific projections are introduced based on the assumed-strain method. The resulting derivation can then be used to model thin structural problems, while taking into account the various through-thickness phenomena. A careful analysis of potential stiffness matrix rank deficiencies reveals that no hourglass modes need to be controlled. However, without assumed-strain method, the element exhibits some shear and thickness-type locking, which is common in linear triangular elements associated with constant strain states. After the formulation of the element is detailed, its performance is assessed through a set of representative benchmark problems illustrating its capabilities in various situations. More specifically, this prismatic solid–shell element proves to be an essential complement to the SHB8PS hexahedral element in meshing arbitrarily complex geometries.

The enhanced assumed strain method for the isogeometric analysis of nearly incompressible deformation of solids

SUMMARYIsogeometric analysis has recently become very popular for the numerical modeling of structures and fluids. Among other potential features, advantages of using a non‐uniform rational B‐splines (NURBS)‐based isogeometric analysis over the traditional finite element method include the possibility of using higher‐order polynomials for the basis functions of the approximation space, which may be easily built on a recursive (hierarchical) fashion as well as higher convergence ratio. Nevertheless, NURBS‐based isogeometric analysis suffers from the same problems depicted by other methods when it comes to reproduce isochoric deformations, that is, it shows volumetric locking, especially for low‐order basis functions. Similar remedies as those that have been proposed for the finite element method may be appropriate for integration in the NURBS‐based isogeometric analysis and some have already been tried with success. In this work, the analysis of the underlying space of incompressible deformations of a NURBS‐based isogeometric approximation is performed with the main objective of understanding the likelihood of volumetric locking. As a remedy, the enhanced assumed strain methodology is blended with the NURBS‐based isogeometric analysis to alleviate the volumetric locking associated with incompressible deformations. The solution includes a stabilization term derived directly from a penalized form of the classical Veubeke–Hu–Washizu three‐field variational principle. Copyright © 2012 John Wiley & Sons, Ltd.

Three‐dimensional meshfree‐enriched finite element formulation for micromechanical hyperelastic modeling of particulate rubber composites

SUMMARYA three‐dimensional microstructure‐based finite element framework is presented for modeling the mechanical response of rubber composites in the microscopic level. This framework introduces a novel finite element formulation, the meshfree‐enriched FEM, to overcome the volumetric locking and pressure oscillation problems that normally arise in the numerical simulation of rubber composites using conventional displacement‐based FEM. The three‐dimensional meshfree‐enriched FEM is composed of five‐noded tetrahedral elements with a volume‐weighted smoothing of deformation gradient between neighboring elements. The L2‐orthogonality property of the smoothing operator enables the employed Hu–Washizu–de Veubeke functional to be degenerated to an assumed strain method, which leads to a displacement‐based formulation that is easily incorporated with the periodic boundary conditions imposed on the unit cell. Two numerical examples are analyzed to demonstrate the effectiveness of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.

Assumed-strain solid–shell formulation for the six-node finite element SHB6: evaluation on non-linear benchmark problems

Because accuracy and efficiency are the main features expected within the finite element (FE) method, the current contribution proposes a six-node prismatic solid–shell, denoted (SHB6). The formulation is extended here to geometric and material nonlinearities, and focus will be placed on its validation on nonlinear benchmark problems. This type of FE is specifically designed for the modeling of thin structures, by combining several useful shell features with some well-known solid element advantages. Therefore, the resulting derivation only involves displacement degrees of freedom as it is based on a fully 3D approach. Some of the motivation behind this formulation is to allow a natural mesh connection in problems where both structural (shell/plate) and continuum (solid) elements need to be simultaneously used. Another major interest of this prismatic solid–shell is to complement meshes that use hexahedral solid–shell FE, especially when free mesh generation tools are employed. To achieve an efficient formulation, the assumed-strain method is combined with an in-plane one-point quadrature scheme. These techniques are intended to reduce both locking phenomena and computational cost. A careful analysis of possible stiffness matrix rank deficiencies demonstrates that this reduced integration procedure does not induce hourglass modes and thus no stabilization is required.

Application of the continuum shell finite element SHB8PS to sheet forming simulation using an extended large strain anisotropic elastic–plastic formulation

This paper proposes an extension of the SHB8PS solid–shell finite element to large strain anisotropic elasto-plasticity, with application to several non-linear benchmark tests including sheet metal forming simulations. This hexahedral linear element has an arbitrary number of integration points distributed along a single line, defining the “thickness” direction; and to control the hourglass modes inherent to this reduced integration, a physical stabilization technique is used. In addition, the assumed strain method is adopted for the elimination of locking. The implementation of the element in Abaqus/Standard via the UEL user subroutine has been assessed through a variety of benchmark problems involving geometric non-linearities, anisotropic plasticity, large deformation and contact. Initially designed for the efficient simulation of elastic–plastic thin structures, the SHB8PS exhibits interesting potentialities for sheet metal forming applications—both in terms of efficiency and accuracy. The element shows good performance on the selected tests, including springback and earing predictions for Numisheet benchmark problems.

A Selective Reduced Integration 8-Node Hexahedral Element by Assumed Strain for Coining Simulation

A selective reduced integration 8-node hexahedral element for coining simulation is developed in this paper. The element is free of volume locking by assumed strain method. The standard velocity gradient matrix is derived in which the shear items are ignored to avoid shear locking. Hourglass modes are successfully suppressed without user-input parameters. The element is successfully employed in the coining simulation package - COINFORM. Numerical tests and experiments of a typical coin are carried out to show the good performances of the element.

New solid‐shell and solid‐beam finite elements with applications to medical technology

AbstractIn this paper, new solid‐shell and solid‐beam finite element formulations for finite deformation problems are introduced. One application of interest, concerning these types of elements, can be found in the simulation of stent implementation in the treatment of stenosis. The beam‐like structure of the stent and the shell‐like structure of the blood vessel can be modelled easily by using these types of elements. Moreover, the modelling of the interaction of the different structural types between each other and with the surounding tissue becomes more simple. A high rate of convergence, by using only one element in thickness direction that is comparable to classical structural elements, can be named as a major requirement for the element formulations. In this regard, different locking effects are cured by a special combination of the assumed natural strain method (ANS) and the enhanced assumed strain method (EAS). In addition a variable number of quadrature points can be used in thickness direction in order to capture nonlinearites. This is combined with the concept of reduced integration for the sake of computational efficiency. An adaptive hourglass stabilization that also accounts for material nonlinearities is a crucial issue in this regard. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A displacement-based nonlinear finite element formulation using meshfree-enriched triangular elements for the two-dimensional large deformation analysis of elastomers

In this paper a displacement-based meshfree-enriched finite element method, which was proposed for the linear modeling of near-incompressible elasticity, is generalized for the nonlinear analysis of elastomers. A four-noded triangular element based on the convex meshfree approximation is utilized to discretize the domain. An area-weighted smoothing of deformation gradient is performed on the four-noded triangular elements to provide a locking-free analysis for near-incompressible materials. The L 2-orthogonality property of the smoothing operator enables the employed Hu–Washizu–de Veubeke variational to be degenerated to an assumed strain method. Different from the extra displacement fields in the conventional assumed strain methods or bubble-enriched finite element methods, the enriched meshfree nodes carry physical quantities and are not eliminated by a local static condensation procedure. This leads to a displacement-based formulation easily to be implemented in the existing finite element code. Several numerical examples are solved to demonstrate the effectiveness and accuracy of the proposed formulation for the large deformation analysis of elastomers.

A new assumed strain solid-shell formulation “SHB6” for the six-node prismatic finite element

This paper presents the development of a new prismatic solid-shell finite element, denoted SHB6, obtained using a purely three-dimensional approach. This element has six nodes with displacements as the only degrees of freedom, and only requires two integration points distributed along a preferential direction, designated as the “thickness”. Although geometrically three-dimensional, this element can be conveniently used to model thin structures while taking into account the various phenomena occurring across the thickness. A reduced integration scheme and specific projections of the strains are introduced, based on the assumed-strain method, in order to improve performance and to eliminate most locking effects. It is first shown that the adopted in-plane reduced integration does not generate “hourglass” modes, but the resulting SHB6 element exhibits some shear and thickness-type locking. This is common in linear triangular elements, in which the strain is constant. The paper details the formulation of this element and illustrates its capabilities through a set of various benchmark problems commonly used in the literature. In particular, it is shown that this new element plays a useful role as a complement to the SHB8PS hexahedral element, which enables one to mesh arbitrary geometries. Examples using both SHB6 and SHB8PS elements demonstrate the advantage of mixing these two solid-shell elements.

Performance of mixed and enhanced finite elements for strain localization in hypoplasticity

SUMMARYDisplacement and mixed finite element formulations of shear localization in materials are presented. The formulations are based on hypoplastic constitutive laws for soils and the mixed enhanced treatment involving displacement, strain and stress rates as independently varied fields. Included in these formulations are the standard displacement method, the three‐field mixed formulation, the enhanced assumed strain method and the mixed enhanced strain method. Several numerical examples demonstrating the capability and performance of the different finite element formulations are presented. The numerical results are compared with available experimental data for Hostun RF sand and numerical results for Karlsruhe sand on biaxial tests. Copyright © 2011 John Wiley & Sons, Ltd.