The patch test conditions and some multivariable finite element formulations

From a unified viewpoint, this article establishes the practical and general constant stress patch test conditions (PTC) for analysing and ensuring convergence or robustness of some multivariable finite element (MFE) formulations, such as hybrid-mixed and incompatible element formulations. The proposed PTC are applied to develop new general MFE formulations which pass PTC and a family of corresponding MFE for stress analysis. Several numerical examples are also given. © 1997 John Wiley & Sons, Ltd.     

Application of the patch test to incompatible laminated finite elements

A new patch test is proposed for laminated finite elements. Two single-element tests are presented which establish the necessary conditions for convergence of incompatible laminated elements. These conditions are applied to a family of incompatible laminated elements recently developed by the authors. A method for applying these conditions to establish a general isoparametric formulation is given. Numerical results from several multielement patch tests performed on these elements are discussed. Additional numerical examples that support the findings of these patch tests are also presented. © 1997 John Wiley & Sons, Ltd.     

Influence of minimum element size to determine crack closure stress by the finite element method

Measuring opening or closure stress is a complex process that influences the low accuracy of obtained data. Finite element models have been one of the available ways to deal with this problem. The difficulty of modelling the whole process of crack growth (due to the great number of cycles implied) as the great complexity of the phenomenon itself (with a high plastic strain concentrated in a small area, with elevated stress gradients) has made the results to be quite varied, being influenced by a great number of modelling parameters. Of those parameters, the minimum size of the element used to mesh the area around the crack tip vicinity presents a great influence on the results.In this work, a detailed analysis of the influence of this parameter in the results in terms of closure or opening stress is presented. The effect that different meshing criteria can have on the result is complex and it has been necessary to reduce the element size around the crack tip to a size that had not been reached before. Procedures and modelling criteria stricter than the ones shown in the current bibliography are proposed. A methodology for the correct interpretation of the results is also established.     

2D finite element analysis of thermally bonded nonwoven materials: Continuous and discontinuous models

Due to a random structure of nonwoven materials, their non-uniform local material properties and nonlinear properties of single fibres, it is difficult to develop a numerical model that adequately accounts for these features and properly describes their performance. Two different finite element (FE) models – continuous and discontinuous – are developed here to describe the tensile behaviour of nonwoven materials. A macro-level continuum finite element model is developed based on the classic composite theory by treating the fibrous network as orthotropic material. This model is used to analyse the effect of thermally bonding points on the deformational behaviour and deformation mechanisms of thermally bonded nonwoven materials at macro-scale. To describe the effects of discontinuous microstructure of the fabric and implement the properties of polypropylene fibres, a micro-level discontinuous finite element model is developed. Applicability of both models to describe various deformational features observed in experiments with a real thermally bonded nonwoven is discussed.     

Three-dimensional finite element analysis of fracture modes for the pull-off test of a thin film from a stiff substrate

A three-dimensional geometrically nonlinear finite element analysis model is presented to study the interfacial delamination for the pull-off test of a thin film strip debonded from a stiff substrate. The strain energy release rates of all three modes (mode I, mode II, and mode III) along the debond front are considered and calculated to investigate the mixed fracture modes for the entire deformation regime from bending plate to stretching membrane. These results indicate that the individual strain energy release rates and the total energy release rate vary along the width of the debond front and strong three-dimensional edge effects exist near the free edges of the film. Interestingly, residual stress also plays an important role in controlling mixed fracture modes and the variation of the energy release rates. Finally, the three-dimensional finite element model is compared with an analytical solution developed earlier. The three-dimensional finite element model is found to provide additional insights for interfacial delamination for the pull-off test.     

Delamination modelling of GLARE using the extended finite element method

In this paper, an application of the Extended Finite Element Method (XFEM) for simulation of delamination in fibre metal laminates is presented. The study consider a double cantilever beam made of fibre metal laminate in which crack opening in mode I and crack propagation were studied. Comparison with the solution by standard Finite Element Method (FEM) as well as with experimental tests is provided. To the authors’ knowledge, this is the first time that XFEM is used in the fracture analysis of fibre metal laminates such as GLARE. The results indicated that XFEM could be a promising technique for the failure analysis of composite structures.     

X-ray tomography applied to the characterization of cellular materials. Related finite element modeling problems

The analyses of several materials exhibiting a cellular structure have been carried out using X-ray tomography. This new technique allows the three dimensional and non destructive visualisation of the studied materials at the scale of their cellular microstructure. Qualitative examples are given for metal foams, bread and cellular concrete. The similarity between these materials is striking. It has been measured by quantitative 3D image processing. The different Finite Element Methods available today to produce meshes from these images are presented and discussed in the final part of this paper.     

Three-dimensional finite element analysis of finite deformation micromorphic linear isotropic elasticity

A finite deformation micromorphic materially linear isotropic elastic model is formulated and implemented for three dimensional finite element analysis. The model is based on the kinematics, balance equations and thermodynamic equations proposed by Eringen and Suhubi (1964). The constitutive equations are calculated in the reference configuration, and the resulting stresses are mapped to the current configuration. The balance of linear momentum and the balance of first moment of momentum are linearized to construct the consistent tangent for three dimensional finite element implementation for solution by the Newton–Raphson method. Three dimensional numerical examples are analyzed to demonstrate preliminarily the implementation.     

A coupled meshfree/finite element method for automotive crashworthiness simulations

Mesh distortion induced numerical instability is a major roadblock in automotive crashworthiness finite element simulations. Remedies such as wrapping elements with null shells and deletion of distorted meshes have been adopted but none of them seems robust enough to survive various scenarios. Meshfree methods have been developed over the past almost twenty years in view of their capabilities in dealing with large material deformation and separation, but have remained in academic research due to their unaffordable high computational cost in solving large-scale industrial applications. This paper presents a coupled meshfree/finite-element method which allows engineers to model the severe deformation area with the meshfree method while keeping the remaining area modeled by the finite element methods. The method is implemented into LS-DYNA version 971 and its later versions so that it is available for automotive crashworthiness simulations. In the paper, one linear patch test and three crash examples are presented to demonstrate the accuracy of the meshfree formulation, its effectiveness in resolving mesh distortion difficulty, and the efficiency of the coupled meshfree/finite element solver in handling large-scale models.     

Evaluation of head response to ballistic helmet impacts using the finite element method

Injuries to the head caused by ballistic impacts are not well understood. Ballistic helmets provide good protection, but still, injuries to both the skull and brain occur. Today there is a lack of relevant test procedure to evaluate the efficiency of a ballistic helmet. The purpose of this project was (1) to study how different helmet shell stiffness affects the load levels in the human head during an impact, and (2) to study how different impact angles affects the load levels in the human head. A detailed finite element (FE) model of the human head, in combination with an FE model of a ballistic helmet (the US Personal Armour System Ground Troops’ (PASGT) geometry) was used. The head model has previously been validated against several impact tests on cadavers. The helmet model was validated against data from shooting tests. Focus was aimed on getting a realistic response of the coupling between the helmet and the head and not on modeling the helmet in detail. The studied data from the FE simulations were stress in the cranial bone, strain in the brain tissue, pressure in the brain, change in rotational velocity and translational and rotational acceleration. A parametric study was performed to see the influence of a variation in helmet shell stiffness on the outputs from the model. The effect of different impact angles was also studied. Dynamic helmet shell deflections larger than the initial distance between the shell and the skull should be avoided in order to protect the head from the most injurious threat levels. It is more likely that a fracture of the skull bone occurs if the inside of the helmet shell strikes the skull. Oblique ballistic impacts may in some cases cause higher strains in the brain tissue than pure radial ones.     

From medical images to anatomically accurate finite element grids

The successful application of computational modelling of blood flow for the planning of surgical and interventional procedures to treat cardiovascular diseases strongly depends on the rapid construction of anatomical models. The large individual variability of the human vasculature and the strong dependence of blood flow characteristics on the vessel geometry require modelling on a patient‐specific basis. Various image processing and geometrical modelling techniques are integrated for the rapid construction of geometrical surface models of arteries starting from medical images. These discretely defined surfaces are then used to generate anatomically accurate finite element grids for hemodynamic simulations. The proposed methodology operates directly in 3D and consists of three stages. In the first stage, the images are filtered to reduce noise and segmented using a region‐growing algorithm in order to obtain a properly defined boundary of the arterial lumen walls. In the second stage, a surface triangulation representing the vessel walls is generated using a direct tessellation of the boundary voxels. This surface is then smoothed and the quality of the resulting triangulation is improved. Finally, in the third stage, the triangulation is subdivided into so‐called discrete surface patches for surface gridding, the desired element size distribution is defined and the finite element grid generated. Copyright © 2001 John Wiley & Sons, Ltd.     

Simulation of the cabling process for Rutherford cables: An advanced finite element model

In all existing large particle accelerators (Tevatron, HERA, RHIC, LHC) the main superconducting magnets are based on Rutherford cables, which are characterized by having: strands fully transposed with respect to the magnetic field, a significant compaction that assures a large engineering critical current density and a geometry that allows efficient winding of the coils. The Nb₃Sn magnets developed in the framework of the HL-LHC project for improving the luminosity of the Large Hadron Collider (LHC) are also based on Rutherford cables. Due to the characteristics of Nb₃Sn wires, the cabling process has become a crucial step in the magnet manufacturing. During cabling the wires experience large plastic deformations that strongly modify the geometrical dimensions of the sub-elements constituting the superconducting strand. These deformations are particularly severe on the cable edges and can result in a significant reduction of the cable critical current as well as of the Residual Resistivity Ratio (RRR) of the stabilizing copper. In order to understand the main parameters that rule the cabling process and their impact on the cable performance, CERN has developed a 3D Finite Element (FE) model based on the LS-Dyna® software that simulates the whole cabling process. In the paper the model is presented together with a comparison between experimental and numerical results for a copper cable produced at CERN.     

Simulation of the effect of material properties and interface roughness on the stress distribution in thermal barrier coatings using finite element method

A Finite Element Model (FEM) was developed to evaluate the stresses induced by the thermal cycling in a typical plasma-sprayed thermal barrier coating system (TBCs). The thermo-mechanical model of this multi-layer system takes into account the effects of thermal and mechanical properties, morphology of the top-coat/bond-coat interface and oxidation on the local stresses that are responsible for the micro-crack nucleation during cooling, especially near the metal/ceramic interface.Two top-coat/bond-coat geometries corresponding to different interfacial asperity morphologies (semicircle or sinusoidal) are modeled considering a two dimensional and periodic geometry. The effect of the geometry and the amplitude of asperities on stress distribution are examined to study the cause of the subsequent delamination of the TBCs system. Moreover, the effect of the creep in all layers and plastic deformation in the bond-coat as well as the oxidation in the perpendicular direction of the top-coat/bond-coat interface are examined toward the stress development and critical sites with respect to possible crack paths. In addition, crack initiation and propagation at the system was predicted.     

Wrinkling of sandwich column: comparison between finite element analysis and analytical solutions

Wrinkling analysis of sandwich columns was carried out. Two methods were used, namely finite element methods and analytical solutions. In the finite element methods, shell elements based on Reissner–Mindlin were employed across the beam thickness. A very fine element was used to model both the skin and the core across the beam thickness. The buckling mode was increased up to 100 in order to be able to model the wrinkling mode. The analytical solutions used energy methods and Raleigh–Ritz solutions developed earlier by the author. Finally, comparisons were made with experimental data taken from published results. Good agreements were achieved between the experimental results and both the finite element and analytical solution. The difference was less than 10%. Both finite element and analytical solutions were able to predict both symmetrical and asymmetrical wrinkling.     

Finite element formulations of strain gradient theory for microstructures and the C0–1 patch test

Based on finite element formulations for the strain gradient theory of microstructures, a convergence criterion for the C^0–1 patch test is introduced, and a new approach to devise strain gradient finite elements that can pass the C^0–1 patch test is proposed. The displacement functions of several plane triangular elements, which satisfy the C⁰ continuity and weak C¹ continuity conditions are evaluated by the C^0–1 patch test. The difference between the proposed C^0–1 patch test and the C⁰ constant stress and C¹ constant curvature patch tests is elucidated. An 18-DOF plane strain gradient triangular element (RCT9+RT9), which passes the C^0–1 patch test and has no spurious zero energy modes, is proposed. Numerical examples are employed to examine the performance of the proposed element by carrying out the C^0–1 patch test and eigenvalue test. The proposed element is found to be without spurious zero energy modes, and it possesses higher accuracy compared with other strain gradient elements. Copyright © 2004 John Wiley & Sons, Ltd.     

Automatic generation of quadrilateral finite element meshes

This paper describes an efficient, accurate and simple implementation of an algorithm for generation of quadrilateral finite element meshes. An original algorithm by Talbert and Parkinson [J.A. Talbert, A. Parkinson, Development of an automatic two-dimensional finite element mesh generator using quadrilateral elements and Bezier curve boundary definition, Int. J. Numer. Meth. Eng., 29 (1990) 1551–1567], has been substantially redeveloped and modified and presented in greater detail. We cover several important issues omitted in publication mentioned and we will provide interested readers with fully documented source code of the program.     

Assessment of a composite beam finite element based on the proper generalized decomposition

In this paper, a finite element based on the Proper Generalized Decomposition (PGD) is presented for the analysis of bi-dimensional laminated beams. The displacement field is approximated as a sum of separated functions of x (axial coordinate) and z (transverse coordinate). This choice yields to an iterative process that consists of computing a product of two one-dimensional functions at each iteration. The capability and the behavior of the PGD approach are shown on isotropic beam with different slenderness ratios. A second and fourth-order expansion with respect to the thickness are considered. Mechanical tests for thin/thick laminated and sandwich beams are presented in order to evaluate the two approaches. They are compared with elasticity and 2D finite element reference solutions.     

Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach

Applications of the Trefftz finite element method to anti-plane electroelastic problems are presented in this paper. A dual variational functional is constructed and used to derive Trefftz finite element formulation. Special trial functions which satisfy boundary conditions are also used to develop a special purpose element with local defects. The performance of the proposed element model is assessed by an example and comparison is made with results obtained by other approaches. The Trefftz finite element approach is demonstrated to be ideally suited for the analysis of the anti-plane problem.The work was performed under the auspices of an Australian Professorial Fellowship Program with grant number DP0209487 and 21st Century Education Promotion Key project from Tianjin University.     

Deformation behavior of a stiffened panel subjected to underwater shock loading using the non-linear finite element method

Given the superior strength-to-weight ratio, stiffened panels have been used extensively in the main structure of ships and underwater vehicles. The loads acting on a stiffened panel in a ship is in-plane compression or tension, resulting from the overall hull-girder bending moment or torsion, shear force resulting from the hull-girder shear force, and lateral pressure resulting from the external wave or shock loading. This work addresses the transient responses of a panel structure reinforced by ribs of different sizes to underwater shock loads using non-linear finite element code-ABAQUS. Verification of the reliability was made between the Ramajeyathilagam’s experiments results [Ramajeyathilagam K, Vendhan CP, Rao VB. Non-linear transient dynamic response of rectangular plates under shock loading. Int J Impact Eng 2000;24:999–1015, Ramajeyathilagam K, Vendhan CP. Deformation and rupture of thin rectangular plates subjected to underwater shock. Int J Impact Eng 2004;30:699–719] at several different locations on the plates. The shock factor is adopted to describe the shock severity. Additionally, the displacement–time histories under different shock loadings are presented which will be used in designing stiffened panels so as to enhance resistance to underwater shock damage.     

An Abaqus implementation of the extended finite element method

In this paper, we introduce an implementation of the extended finite element method for fracture problems within the finite element software ABAQUS^TM. User subroutine (UEL) in Abaqus is used to enable the incorporation of extended finite element capabilities. We provide details on the data input format together with the proposed user element subroutine, which constitutes the core of the finite element analysis; however, pre-processing tools that are necessary for an X-FEM implementation, but not directly related to Abaqus, are not provided. In addition to problems in linear elastic fracture mechanics, non-linear frictional contact analyses are also realized. Several numerical examples in fracture mechanics are presented to demonstrate the benefits of the proposed implementation.