Adaptive dynamic cohesive fracture simulation using nodal perturbation and edge‐swap operators

Dependence on mesh orientation impacts adversely the quality of computational solutions generated by cohesive zone models. For instance, when considering crack propagation along interfaces between finite elements of 4k structured meshes, both extension of crack length and crack angle are biased according to the mesh configuration. To address mesh orientation dependence in 4k structured meshes and to avoid undesirable crack patterns, we propose the use of nodal perturbation (NP) and edge‐swap (ES) topological operation. To this effect, the topological data structure TopS (Int. J. Numer. Meth. Engng 2005; 64 : 1529–1556), based on topological entities (node, element, vertex, edge and facet), is utilized so that it is possible to access adjacency information and to manage a consistent data structure in time proportional to the number of retrieved entities. In particular, the data structure allows the ES operation to be done in constant time. Three representative dynamic fracture examples using ES and NP operators are provided: crack propagation in the compact compression specimen, local branching instability, and fragmentation. These examples illustrate the features of the present computational framework in simulating a range of physical phenomena associated with cracking. Copyright © 2010 John Wiley & Sons, Ltd.     

Mathematical model for the noise–lifetime relationship and its application to reliability screening

This paper puts forward a two‐dimensional probability distribution. The first ‘wing’ of this distribution is the device lifetime. The second ‘wing’ is the most promising reliability indicator, namely the 1/f noise factor. The model is intended to serve as a basis for reliability screening. It involves the noise–reliability correlation coefficient and has an attractive engineering interpretation. It ascribes a two‐element series reliability structure to the device. The first element of this structure is noise‐independent and ‘real’ in the sense that its hazard rate is always of positive value. The second element is noise‐dependent. The element is imaginary in the sense that its hazard rate can be changed from positive to negative value as the noise–reliability correlation coefficient changes in the same direction. Copyright © 2000 John Wiley & Sons, Ltd.     

Nondestructive identification of multiple flaws using XFEM and a topologically adapting artificial bee colony algorithm

We present a novel algorithm based on the extended finite element method (XFEM) and an enhanced artificial bee colony (EABC) algorithm to detect and quantify multiple flaws in structures. The concept is based on recent work that have shown the excellent synergy between XFEM, used to model the forward problem, and a genetic‐type algorithm to solve an inverse identification problem and converge to the ‘best’ flaw parameters. In this paper, an adaptive algorithm that can detect multiple flaws without any knowledge on the number of flaws beforehand is proposed. The algorithm is based on the introduction of topological variables into the search space, used to adaptively activate/deactivate flaws during run time until convergence is reached. The identification is based on a limited number of strain sensors assumed to be attached to the structure surface boundaries. Each flaw is approximated by a circular void with the following three variables: center coordinates (x_c, y_c) and radius (r_c), within the XFEM framework. In addition, the proposed EABC scheme is improved by a guided‐to‐best solution updating strategy and a local search (LS) operator of the Nelder–Mead simplex type that show fast convergence and superior global/LS abilities compared with the standard ABC or classic genetic algorithms. Several numerical examples, with increasing level of difficulty, are studied in order to evaluate the proposed algorithm. In particular, we consider identification of multiple flaws with unknown a priori information on the number of flaws (which makes the inverse problem harder), the proximity of flaws, flaws having irregular shapes (similar to artificial noise), and the effect of structured/unstructured meshes. The results show that the proposed XFEM–EABC algorithm is able to converge on all test problems and accurately identify flaws. Hence, this methodology is found to be robust and efficient for nondestructive detection and quantification of multiple flaws in structures. Copyright © 2013 John Wiley & Sons, Ltd.     

The application of node-element rules for forecasting problems in the generation of finite element meshes

The number of equations which relate the mesh parameters (number of nodes, number of elements etc.) to each other is smaller than the number of unknowns contained therein. These rules are only applicable when several parameters of a defined mesh are known, but are unsuitable for use in the particularly interesting field of mesh projection unless additional inter-relationships can be found. Such relationships are given for uniform meshes in which besides the characteristics of the element type the ‘solidity value’ of the structure plays an important role. The equations are derived for two-dimensional triangular and quadrilateral elements and also for three-dimensional tetrahedral and hexahedral elements. The fact that these relationships are also suitable for estimating the parameters of any non-uniform mesh, discloses a wide field of application in the management of storage and in the control of programs for automatic mesh generation.     

Extrinsic cohesive modelling of dynamic fracture and microbranching instability in brittle materials

Dynamic crack microbranching processes in brittle materials are investigated by means of a computational fracture mechanics approach using the finite element method with special interface elements and a topological data structure representation. Experiments indicate presence of a limiting crack speed for dynamic crack in brittle materials as well as increasing fracture resistance with crack speed. These phenomena are numerically investigated by means of a cohesive zone model (CZM) to characterize the fracture process. A critical evaluation of intrinsic versus extrinsic CZMs is briefly presented, which highlights the necessity of adopting an extrinsic approach in the current analysis. A novel topology‐based data structure is employed to enable fast and robust manipulation of evolving mesh information when extrinsic cohesive elements are inserted adaptively. Compared to intrinsic CZMs, which include an initial hardening segment in the traction–separation curve, extrinsic CZMs involve additional issues both in implementing the procedure and in interpreting simulation results. These include time discontinuity in stress history, fracture pattern dependence on time step control, and numerical energy balance. These issues are investigated in detail through a ‘quasi‐steady‐state’ crack propagation problem in polymethylmethacrylate. The simulation results compare reasonably well with experimental observations both globally and locally, and demonstrate certain advantageous features of the extrinsic CZM with respect to the intrinsic CZM. Copyright © 2007 John Wiley & Sons, Ltd.     

An unsymmetric 8‐node hexahedral element with high distortion tolerance

Among all 3D 8‐node hexahedral solid elements in current finite element library, the ‘best’ one can produce good results for bending problems using coarse regular meshes. However, once the mesh is distorted, the accuracy will drop dramatically. And how to solve this problem is still a challenge that remains outstanding. This paper develops an 8‐node, 24‐DOF (three conventional DOFs per node) hexahedral element based on the virtual work principle, in which two different sets of displacement fields are employed simultaneously to formulate an unsymmetric element stiffness matrix. The first set simply utilizes the formulations of the traditional 8‐node trilinear isoparametric element, while the second set mainly employs the analytical trial functions in terms of 3D oblique coordinates (R, S, T). The resulting element, denoted by US‐ATFH8, contains no adjustable factor and can be used for both isotropic and anisotropic cases. Numerical examples show it can strictly pass both the first‐order (constant stress/strain) patch test and the second‐order patch test for pure bending, remove the volume locking, and provide the invariance for coordinate rotation. Especially, it is insensitive to various severe mesh distortions. Copyright © 2016 John Wiley & Sons, Ltd.     

A family of second‐order boundary dampers for finite element analysis of two and three‐dimensional problems in transient exterior acoustics

Numerical modelling of exterior acoustics problems involving infinite medium requires truncation of the medium at a finite distance from the obstacle or the structure and use of non‐reflecting boundary condition at this truncation surface to simulate the asymptotic behaviour of radiated waves at far field. In the context of the finite element method, Bayliss–Gunzburger–Turkel (BGT) boundary conditions are well suited since they are local in both space and time. These conditions involve ‘damper’ operators of various orders, which work on acoustic pressure p and they have been used in time harmonic problems widely and in transient problems in a limited way. Alternative forms of second‐order BGT operators, which work on ṗ (time derivative of p) had been suggested in an earlier paper for 3D problems but they were neither implemented nor validated. This paper presents detailed formulations of these second‐order dampers both for 2D and 3D problems, implements them in a finite element code and validates them using appropriate example problems. The developed code is capable of handling exterior acoustics problems involving both Dirichlet and Neumann boundary conditions. Copyright © 2000 John Wiley & Sons, Ltd.     

Using the sequential linear integer programming method as a post‐processor for stress‐constrained topology optimization problems

This paper deals with topology optimization of load‐carrying structures defined on discretized continuum design domains. In particular, the minimum compliance problem with stress constraints is considered. The finite element method is used to discretize the design domain into n finite elements and the design of a certain structure is represented by an n‐dimensional binary design variable vector. In order to solve the problems, the binary constraints on the design variables are initially relaxed and the problems are solved with both the method of moving asymptotes and the sparse non‐linear optimizer solvers for continuous optimization in order to compare the two solvers. By solving a sequence of problems with a sequentially lower limit on the amount of grey allowed, designs that are close to ‘black‐and‐white’ are obtained. In order to get locally optimal solutions that are purely {0, 1}ⁿ, a sequential linear integer programming method is applied as a post‐processor. Numerical results are presented for some different test problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Fracture toughness and growth of short and long fatigue cracks in ductile cast iron EN‐GJS‐400‐18‐LT

Heavy components of ductile cast iron frequently exhibit metallurgical defects that behave like cracks under cyclic loading. Thus, in order to decide whether a given defect is permissible, it is important to establish the fatigue crack growth properties of the material. In this paper, results from a comprehensive study of ductile cast iron EN‐GJS‐400‐18‐LT have been reported. Growth rates of fatigue cracks ranging from a few tenths of a millimetre (‘short’ cracks) to several millimetres (‘long’ cracks) have been measured for load ratios R=?1, R= 0 and R= 0.5 using a highly sensitive potential‐drop technique. Short cracks were observed to grow faster than long cracks. The threshold stress intensity range, ΔK_th, as a function of the load ratio was fitted to a simple crack closure model. Fatigue crack growth data were compared with data from other laboratories. Single plain fatigue tests at R=?1 and R= 0 were also carried out. Fracture toughness was measured at temperatures ranging from ?40 °C to room temperature.     

A vectorized version of a sparse matrix-vector multiply

A fast vectorized algorithm is presented for a sparse matrix-vector multiply. It can be used when the matrix, A , can be represented as a multiplitting, A = ∑ A _e. In particular, it can be applied to a matrix-vector multiply arising in finite element techniques where the matrices A _e are associated with the individual element contributions to the global matrix A . The algorithm presented here uses a data structure which is based on the individual matrices A _e and can be applied both to symmetric and to non-symmetric matrices. This algorithm would be attractive for vector architecture similar to either the CYBER 205 or the CRAY and has been implemented for both regular and irregular finite element grids on the CYBER 205. Execution times and storage requirements are compared to standard sparse and band matrix-vector multiply algorithms.     

Scaled boundary finite‐element analysis of a non‐homogeneous elastic half‐space

As a result of stresses experienced during and after the deposition phase, a soil strata of uniform material generally exhibits an increase in elastic stiffness with depth. The immediate settlement of foundations on deep soil deposits and the resultant stress state within the soil mass may be most accurately calculated if this increase in stiffness with depth is taken into account. This paper presents an axisymmetric formulation of the scaled boundary finite‐element method and incorporates non‐homogeneous elasticity into the method. The variation of Young's modulus (E) with depth (z) is assumed to take the form E=m_Ez^α, where m_E is a constant and αis the non‐homogeneity parameter. Results are presented and compared to analytical solutions for the settlement profiles of rigid and flexible circular footings on an elastic half‐space, under pure vertical load with αvarying between zero and one, and an example demonstrating the versatility and practicality of the method is also presented. Known analytical solutions are accurately represented and new insight regarding displacement fields in a non‐homogeneous elastic half‐space is gained. Copyright © 2003 John Wiley & Sons, Ltd.     

Conformal higher‐order remeshing schemes for implicitly defined interface problems

A new higher‐order accurate method is proposed that combines the advantages of the classical p‐version of the FEM on body‐fitted meshes with embedded domain methods. A background mesh composed by higher‐order Lagrange elements is used. Boundaries and interfaces are described implicitly by the level set method and are within elements. In the elements cut by the boundaries or interfaces, an automatic decomposition into higher‐order accurate sub‐elements is realized. Therefore, the zero level sets are detected and meshed in a first step, which is called reconstruction. Then, based on the topological situation in the cut element, higher‐order sub‐elements are mapped to the two sides of the boundary or interface. The quality of the reconstruction and the mapping largely determines the properties of the resulting, automatically generated conforming mesh. It is found that optimal convergence rates are possible although the resulting sub‐elements are not always well‐shaped. Copyright © 2016 John Wiley & Sons, Ltd.     

Mapping Cohesive Fracture and Fragmentation Simulations to Graphics Processor Units

A graphics processor units(GPU)‐based computational framework is presented to deal with dynamic failure events simulated by means of cohesive zone elements. The work is divided into two parts. In the first part, we deal with pre‐processing of the information and verify the effectiveness of dynamic insertion of cohesive elements in large meshes in parallel. To this effect, we employ a novel and simplified topological data structure specialized for meshes with triangles, designed to run efficiently and minimize memory occupancy on the GPU. In the second part, we present a parallel explicit dynamics code that implements an extrinsic cohesive zone formulation where the elements are inserted ‘on‐the‐fly’, when needed and where needed. The main challenge for implementing a GPU‐based computational framework using an extrinsic cohesive zone formulation resides on being able to dynamically adapt the mesh, in a consistent way, by inserting cohesive elements on fractured facets. In order to handle that, we extend the conventional data structure used in the finite element method (based on element incidence) and store, for each element, references to the adjacent elements. This additional information suffices to consistently insert cohesive elements by duplicating nodes when needed. Currently, our data structure is specialized for triangular meshes, but an extension to tetrahedral meshes is feasible. The data structure is effective when used in conjunction with algorithms to traverse nodes and elements. Results from parallel simulations show an increase in performance when adopting strategies such as distributing different jobs among threads for the same element and launching many threads per element. To avoid concurrency on accessing shared entities, we employ graph coloring. In a pre‐processing phase, each node of the dual graph (bulk elements of the mesh as graph nodes) is assigned a color different from the colors assigned to adjacent nodes. In that fashion, elements of the same color can be processed in parallel without concurrency. All the procedures needed for the insertion of cohesive elements along fracture facets and for computing nodal properties are performed by threads assigned to triangles, invoking one kernel per color. Computations on existing cohesive elements are also performed based on adjacent bulk elements. Experiments show that GPU speedup increases with the number of nodes and bulk elements. Copyright © 2015 John Wiley & Sons, Ltd.     

Non‐linear exact geometry 12‐node solid‐shell element with three translational degrees of freedom per node

This paper presents the finite rotation exact geometry (EG) 12‐node solid‐shell element with 36 displacement degrees of freedom. The term ‘EG’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at each element node. The finite element formulation developed is based on the 9‐parameter shell model by employing a new concept of sampling surfaces (S‐surfaces) inside the shell body. We introduce three S‐surfaces, namely, bottom, middle and top, and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows one to represent the finite rotation higher order EG solid‐shell element formulation in a very compact form and to derive the strain–displacement relationships, which are objective, that is, invariant under arbitrarily large rigid‐body shell motions in convected curvilinear coordinates. The tangent stiffness matrix is evaluated by using 3D analytical integration and the explicit presentation of this matrix is given. The latter is unusual for the non‐linear EG shell element formulation. Copyright © 2011 John Wiley & Sons, Ltd.     

Discovery of Superconductivity in 2M WS2 with Possible Topological Surface States

Recently the metastable 1T′‐type VIB‐group transition metal dichalcogenides (TMDs) have attracted extensive attention due to their rich and intriguing physical properties, including superconductivity, valleytronics physics, and topological physics. Here, a new layered WS₂ dubbed “2M” WS₂, is constructed from 1T′ WS₂ monolayers, is synthesized. Its phase is defined as 2M based on the number of layers in each unit cell and the subordinate crystallographic system. Intrinsic superconductivity is observed in 2M WS₂ with a transition temperature T_c of 8.8 K, which is the highest among TMDs not subject to any fine‐tuning process. Furthermore, the electronic structure of 2M WS₂ is found by Shubnikov–de Haas oscillations and first‐principles calculations to have a strong anisotropy. In addition, topological surface states with a single Dirac cone, protected by topological invariant Z₂, are predicted through first‐principles calculations. These findings reveal that the new 2M WS₂ might be an interesting topological superconductor candidate from the VIB‐group transition metal dichalcogenides.     

Moving particle finite element method with global smoothness

We describe a new version of the moving particle finite element method (MPFEM) that provides solutions within a C⁰ finite element framework. The finite elements determine the weighting for the moving partition of unity. A concept of ‘General Shape Function’ is proposed which extends regular finite element shape functions to a larger domain. These are combined with Shepard functions to obtain a smooth approximation. The Moving Particle Finite Element Method combines desirable features of finite element and meshfree methods. The proposed approach, in fact, can be interpreted as a ‘moving partition of unity finite element method’ or ‘moving kernel finite element method’. This method possesses the robustness and efficiency of the C⁰ finite element method while providing at least C¹ continuity. Copyright © 2004 John Wiley & Sons, Ltd.     

Improved implementation and robustness study of the X‐FEM for stress analysis around cracks

Numerical crack propagation schemes were augmented in an elegant manner by the X‐FEM method. The use of special tip enrichment functions, as well as a discontinuous function along the sides of the crack allows one to do a complete crack analysis virtually without modifying the underlying mesh, which is of industrial interest, especially when a numerical model for crack propagation is desired. This paper improves the implementation of the X‐FEM method for stress analysis around cracks in three ways. First, the enrichment strategy is revisited. The conventional approach uses a ‘topological’ enrichment (only the elements touching the front are enriched). We suggest a ‘geometrical’ enrichment in which a given domain size is enriched. The improvements obtained with this enrichment are discussed. Second, the conditioning of the X‐FEM both for topological and geometrical enrichments is studied. A preconditioner is introduced so that ‘off the shelf’ iterative solver packages can be used and perform as well on X‐FEM matrices as on standard FEM matrices. The preconditioner uses a local (nodal) Cholesky based decomposition. Third, the numerical integration scheme to build the X‐FEM stiffness matrix is dramatically improved for tip enrichment functions by the use of an ad hoc integration scheme. A 2D benchmark problem is designed to show the improvements and the robustness. Copyright © 2005 John Wiley & Sons, Ltd.     

Isogeometric finite element data structures based on Bézier extraction of T‐splines

We develop finite element data structures for T‐splines based on Bézier extraction generalizing our previous work for NURBS. As in traditional finite element analysis, the extracted Bézier elements are defined in terms of a fixed set of polynomial basis functions, the so‐called Bernstein basis. The Bézier elements may be processed in the same way as in a standard finite element computer program, utilizing exactly the same data processing arrays. In fact, only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same. A byproduct of the extraction process is the element extraction operator. This operator localizes the topological and global smoothness information to the element level, and represents a canonical treatment of T‐junctions, referred to as ‘hanging nodes’ in finite element analysis and a fundamental feature of T‐splines. A detailed example is presented to illustrate the ideas. Copyright © 2011 John Wiley & Sons, Ltd.     

Developing a model‐based sampling plan

Statistical process modeling is widely used in industry for forecasting the production outcomes, for process control and for process optimization. Applying a prediction model in a production process allows the user to calibrate/predict the mean of the distribution of the process outcomes and to partition the overall variation in the distribution of the process outcomes into explained (by the model) and unexplained (residuals) variations; thus, reducing the unexplained variability. The additional information about the process behavior can be used prior to the sampling procedure and may help to reduce the required sample size to classify a lot. This research focuses on the development of a model‐based sampling plan based ontextitCpk (process capability index). It is an extension of a multistage acceptance sampling plan also based on Cpk (Negrin et al., Quality Engineering 2009; 21 :306–318; Quality and Reliability Engineering International 2011; 27 :3–14). The advantage of this sampling plan is that the sample size needed depends directly and quantitatively on the quality of the process (Cpk), whereas other sampling plans such as MIL‐STD‐414 (Sampling Procedures and Tables for Inspection by Variables for Percent Defective, Department of Defense, Washington, DC, 1957.) use only qualitative measures. The objective of this paper is to further refine the needed sample size by using a predictive model for the lot's expectation. We developed model‐based sample size formulae which depend directly on the quality of the prediction model (as measured by R²) and adjust the ‘not model‐based’ multistage sampling plan developed in Negrin et al. (Quality Engineering 2009; 21 :306–318; Quality and Reliability Engineering International 2011; 27 :3–14) accordingly. A simulation study was conducted to compare between the model‐based and the ‘not model‐based’ sampling plans. It is found that when R² = 0, the model‐based and ‘not model‐based’ sampling plans require the same sample sizes in order to classify the lots. However, as R² becomes larger, the sample size required by the model‐based sampling plan becomes smaller than the one required by the ‘not model‐based’ sampling plan. In addition, it is found that the reduction of the sample size achieved by the model‐based sampling plan becomes more significant as Cpk tends to 1 and can be achieved without increasing the proportion of the classification errors. Finally, the suggested sampling plan was applied with areal data set from a chemicals manufacturing process for illustration. Copyright © 2011 John Wiley & Sons, Ltd.     

A convenient way to represent fatigue crack growth in structural adhesives

The present paper examines crack growth in a range of aerospace and automotive structural adhesive joints under cyclic‐fatigue loadings. It is shown that cyclic‐fatigue crack growth in such materials can be represented by a form of the Hartman–Schijve crack‐growth equation, which aims to give a unique and linear ‘master’ representation for the fatigue data points that have been experimentally obtained, as well as enabling the basic fatigue relationship to be readily computed. This relationship is shown to capture the experimental data representing the effects of test conditions, such as R‐ratio and test temperature. It also captures the typical scatter often seen in the fatigue crack‐growth tests, especially at low values of the fatigue crack‐growth rate. The methodology is also shown to be applicable to both Mode I (opening tensile), Mode II (in‐plane shear) and Mixed‐Mode I/II fatigue loadings. Indeed, it has been demonstrated that the fatigue behaviour of structural adhesives under both Mode I and Mode II loadings may be described by one unique ‘master’ linear relationship via the Hartman–Schijve approach.