A high‐order generalized FEM for through‐the‐thickness branched cracks

This paper presents high‐order implementations of a generalized finite element method for through‐the‐thickness three‐dimensional branched cracks. This approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks, independently of the background finite element mesh. Representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies. The combination of local refinement at crack fronts and high‐order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

Mesh refinement strategies without mapping of nonlinear solutions for the generalized and standard FEM analysis of 3‐D cohesive fractures

A robust and efficient strategy is proposed to simulate mechanical problems involving cohesive fractures. This class of problems is characterized by a global structural behavior that is strongly affected by localized nonlinearities at relatively small‐sized critical regions. The proposed approach is based on the division of a simulation into a suitable number of sub‐simulations where adaptive mesh refinement is performed only once based on refinement window(s) around crack front process zone(s). The initialization of Newton‐Raphson nonlinear iterations at the start of each sub‐simulation is accomplished by solving a linear problem based on a secant stiffness, rather than a volume mapping of nonlinear solutions between meshes. The secant stiffness is evaluated using material state information stored/read on crack surface facets which are employed to explicitly represent the geometry of the discontinuity surface independently of the volume mesh within the generalized finite element method framework. Moreover, a simplified version of the algorithm is proposed for its straightforward implementation into existing commercial software. Data transfer between sub‐simulations is not required in the simplified strategy. The computational efficiency, accuracy, and robustness of the proposed strategies are demonstrated by an application to cohesive fracture simulations in 3‐D. Copyright © 2016 John Wiley & Sons, Ltd.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

An h‐hierarchical adaptive procedure for the scaled boundary finite‐element method

The scaled boundary finite‐element method (a novel semi‐analytical method for solving linear partial differential equations) involves the solution of a quadratic eigenproblem, the computational expense of which rises rapidly as the number of degrees of freedom increases. Consequently, it is desirable to use the minimum number of degrees of freedom necessary to achieve the accuracy desired. Stress recovery and error estimation techniques for the method have recently been developed. This paper describes an h‐hierarchical adaptive procedure for the scaled boundary finite‐element method. To allow full advantage to be taken of the ability of the scaled boundary finite‐element method to model stress singularities at the scaling centre, and to avoid discretization of certain adjacent segments of the boundary, a sub‐structuring technique is used. The effectiveness of the procedure is demonstrated through a set of examples. The procedure is compared with a similar h‐hierarchical finite element procedure. Since the error estimators in both cases evaluate the energy norm of the stress error, the computational cost of solutions of similar overall accuracy can be compared directly. The examples include the first reported direct comparison of the computational efficiency of the scaled boundary finite‐element method and the finite element method. The scaled boundary finite‐element method is found to reduce the computational effort considerably. Copyright © 2002 John Wiley & Sons, Ltd.     

Efficient dynamic finite element method for 3D objects with uniform cross‐section

An efficient implicit dynamic finite element method (FEM) for elastic 3D objects with uniform cross‐sections was developed. In this method, the finite element mesh is generated in such a way that the object to be analysed is at first sliced into layers with the same thickness along its generatrix and then each layer is discretized into finite elements of the same pattern. This way of discretization makes the mass, viscosity, and stiffness matrices into the repetitive block tridiagonal matrices. The repetitive block tridiagonal matrix has the characteristic, that the sequence of matrices which appears in the Gaussian elimination for the repetitive block tridiagonal matrix is a rapid convergent sequence. The process of the Gaussian elimination can be terminated when the sequence converges. The rest of the sequence is not necessary to be stored. The present method can save the computational time and memory by utilising this characteristic of the repetitive block tridiagonal matrix. A few examples of analyses including whole Hopkinson‐bar analysis were performed to demonstrate the effectiveness of the present method. The present method is applicable not only to the elasto‐dynamics but also to many other problems, such as thermal problems, electrical problems, and plastic problems without geometric non‐linearity. Copyright © 2005 John Wiley & Sons, Ltd.     

Adaptive sparse grid based HOPGD: Toward a nonintrusive strategy for constructing space‐time welding computational vademecum

Simulation‐based engineering usually needs the construction of computational vademecum to take into account the multiparametric aspect. One example concerns the optimization and inverse identification problems encountered in welding processes. This paper presents a nonintrusive a posteriori strategy for constructing quasi‐optimal space‐time computational vademecum using the higher‐order proper generalized decomposition method. Contrary to conventional tensor decomposition methods, based on full grids (eg, parallel factor analysis/higher‐order singular value decomposition), the proposed method is adapted to sparse grids, which allows an efficient adaptive sampling in the multidimensional parameter space. In addition, a residual‐based accelerator is proposed to accelerate the higher‐order proper generalized decomposition procedure for the optimal aspect of computational vademecum. Based on a simplified welding model, different examples of computational vademecum of dimension up to 6, taking into account both geometry and material parameters, are presented. These vademecums lead to real‐time parametric solutions and can serve as handbook for engineers to deal with optimization, identification, or other problems related to repetitive task.     

Generalized B‐bar: unlocking multiple compliance modes of finite elements with stiff‐fiber anisotropy

Element locking is often seen in homogenized models of elastic fiber‐reinforced materials, and splitting the material compliance into two separate terms isolates troublesome strain modes. Once isolated, the locking modes can be addressed with tailored integration schemes or the opportune introduction of field variables. The canonical application of this approach is seen in the dilatational‐deviatoric split used to treat so‐called ‘volumetric locking’. In the present work, we invoke the spectral decomposition of the material compliance to provide a generalized split. Doing so naturally parses the response into six independent strain modes, with varying propensity for locking. This split can be used to generalize fundamental techniques, such as selective reduced integration and the B‐bar method. This broadened approach works to remedy locking suffered by lower order finite elements used to discretize troublesome materials. Applying these generalized methods to achieve the dilational‐deviatoric split is trivial. However, the compliance spectrum's ability to naturally isolate stiff material response modes makes it a uniquely valuable tool for use on homogenized anisotropic materials. Applying the split, defined by only the first compliance mode, has given rise to the generalized methods, which have proven effective in unlocking finite element models of anisotropic materials. In the present work, the generalization is broadened to treat more than one constrained mode. While treating six modes is equivalent to simple reduced integration techniques, up to five compliance modes are now separated for advantageous treatment. However, some attention must be paid to the stability of the resulting finite element stiffness matrices. We focus here on the treatment of two principal compliance modes. These ‘two‐mode’ applications of the generalized B‐bar method are shown to be a more robust default treatment of linear hexahedral elements than is provided by classical selective reduced integration. This is achieved with a negligible computational overhead. A framework for assessing element stability is delineated, and commonly arising instabilities are analyzed. Copyright © 2016 John Wiley & Sons, Ltd.     

Parallel computing of high‐speed compressible flows using a node‐based finite‐element method

An efficient parallel computing method for high‐speed compressible flows is presented. The numerical analysis of flows with shocks requires very fine computational grids and grid generation requires a great deal of time. In the proposed method, all computational procedures, from the mesh generation to the solution of a system of equations, can be performed seamlessly in parallel in terms of nodes. Local finite‐element mesh is generated robustly around each node, even for severe boundary shapes such as cracks. The algorithm and the data structure of finite‐element calculation are based on nodes, and parallel computing is realized by dividing a system of equations by the row of the global coefficient matrix. The inter‐processor communication is minimized by renumbering the nodal identification number using ParMETIS. The numerical scheme for high‐speed compressible flows is based on the two‐step Taylor–Galerkin method. The proposed method is implemented on distributed memory systems, such as an Alpha PC cluster, and a parallel supercomputer, Hitachi SR8000. The performance of the method is illustrated by the computation of supersonic flows over a forward facing step. The numerical examples show that crisp shocks are effectively computed on multiprocessors at high efficiency. Copyright © 2003 John Wiley & Sons, Ltd.     

Lower‐dimensional interface elements with local enrichment: application to coupled hydro‐mechanical problems in discretely fractured porous media

In this study, we develop lower‐dimensional interface elements to represent preexisting fractures in rock material, focusing on finite element analysis of coupled hydro‐mechanical problems in discrete fractures–porous media systems. The method adopts local enrichment approximations for a discontinuous displacement and a fracture relative displacement function. Multiple and intersected fractures can be treated with the new scheme. Moreover, the method requires less mesh dependencies for accurate finiteelement approximations compared with the conventional interface element method. In particular, for coupled problems, the method allows for the use of a single mesh for both mechanical and other related processes such as flow and transport. For verification purposes, several numerical examples are examined in detail. Application to a coupled hydro‐mechanical problem is demonstrated with fluid injection into a single fracture. The numerical examples prove that the proposed method produces results in strong agreement with reference solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto‐viscoplastic plane‐strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto‐viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto‐plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non‐hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.     

The extended/generalized finite element method: An overview of the method and its applications

An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non‐smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non‐smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi‐field problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Mesh‐independent p‐orthotropic enrichment using the generalized finite element method

This paper is aimed at presenting a simple yet effective procedure to implement a mesh‐independent p‐orthotropic enrichment in the generalized finite element method. The procedure is based on the observation that shape functions used in the GFEM can be constructed from polynomials defined in any co‐ordinate system regardless of the underlying mesh or type of element used. Numerical examples where the solution possesses boundary or internal layers are solved on coarse tetrahedral meshes with isotropic and the proposed p‐orthotropic enrichment. Copyright © 2002 John Wiley & Sons, Ltd.     

Coupling finite element and reliability analysis through proper generalized decomposition model reduction

The FEM is the main tool used for structural analysis. When the design of the mechanical system involves uncertain parameters, a coupling of the FEM with reliability analysis algorithms allows to compute the failure probability of the system. However, this coupling leads to successive finite element analysis of parametric models involving high computational effort. Over the past years, model reduction techniques have been developed in order to reduce the computational requirements in the numerical simulation of complex models. The objective of this work is to propose an efficient methodology to compute the failure probability for a multi‐material elastic structure, where the Young moduli are considered as uncertain variables. A proper generalized decomposition algorithm is developed to compute the solution of parametric multi‐material model. This parametrized solution is used in conjunction with a first‐order reliability method to compute the failure probability of the structure. Applications to multilayered structures in two‐dimensional plane elasticity are presented.Copyright © 2013 John Wiley & Sons, Ltd.     

On semi‐analytical probabilistic finite element method for homogenization of the periodic fiber‐reinforced composites

The main aim of this paper is a development of the semi‐analytical probabilistic version of the finite element method (FEM) related to the homogenization problem. This approach is based on the global version of the response function method and symbolic integral calculation of basic probabilistic moments of the homogenized tensor and is applied in conjunction with the effective modules method. It originates from the generalized stochastic perturbation‐based FEM, where Taylor expansion with random parameters is not necessary now and is simply replaced with the integration of the response functions. The hybrid computational implementation of the system MAPLE with homogenization‐oriented FEM code MCCEFF is invented to provide probabilistic analysis of the homogenized elasticity tensor for the periodic fiber‐reinforced composites. Although numerical illustration deals with a homogenization of a composite with material properties defined as Gaussian random variables, other composite parameters as well as other probabilistic distributions may be taken into account. The methodology is independent of the boundary value problem considered and may be useful for general numerical solutions using finite or boundary elements, finite differences or volumes as well as for meshless numerical strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

Anomalous behavior of bilinear quadrilateral finite elements for modeling cohesive cracks with XFEM/GFEM

The performance of partition‐of‐unity based methods such as the generalized finite element method or the extended finite element method is studied for the simulation of cohesive cracking. The focus of investigation is on the performance of bilinear quadrilateral finite elements using these methods. In particular, the approximation of the displacement jump field, representing cohesive cracks, by extended finite element method/generalized finite element method and its effect on the overall behavior at element and structural level is investigated. A single element test is performed with two different integration schemes, namely the Newton‐Cotes/Lobatto and the Gauss integration schemes, for the cracked interface contribution. It was found that cohesive crack segments subjected to a nonuniform opening in unstructured meshes (or an inclined crack in a structured finite element mesh) result in an unrealistic crack opening. The reasons for such behavior and its effect on the response at element level are discussed. Furthermore, a mesh refinement study is performed to analyze the overall response of a cohesively cracked body in a finite element analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Boundary elements for half‐space problems via fundamental solutions: A three‐dimensional analysis

An efficient solution technique is proposed for the three‐dimensional boundary element modelling of half‐space problems. The proposed technique uses alternative fundamental solutions of the half‐space (Mindlin's solutions for isotropic case) and full‐space (Kelvin's solutions) problems. Three‐dimensional infinite boundary elements are frequently employed when the stresses at the internal points are required to be evaluated. In contrast to the published works, the strongly singular line integrals are avoided in the proposed solution technique, while the discretization of infinite elements is independent of the finite boundary elements. This algorithm also leads to a better numerical accuracy while the computational time is reduced. Illustrative numerical examples for typical isotropic and transversely isotropichalf‐space problems demonstrate the potential applications of the proposed formulations. Incidentally, the results of the illustrative examples also provide a parametric study for the imperfect contact problem. Copyright © 2001 John Wiley & Sons, Ltd.     

A kernel free particle‐finite element method for hypervelocity impact simulation

An improved hybrid particle‐finite element method has been developed for the simulation of hypervelocity impact problems. Unlike alternative methods, the revised formulation computes the density without reference to any kernel or interpolation functions, for either the density or the rate of dilatation. This simplifies the state space model and leads to a significant reduction in computational cost. The improved method introduces internal energy variables as generalized co‐ordinates in a new formulation of the thermomechanical Lagrange equations. Example problems show good agreement with exact solutions in one dimension and good agreement with experimental data in a three‐dimensional simulation. Copyright © 2005 John Wiley & Sons, Ltd.     

Use of higher‐order shape functions in the scaled boundary finite element method

The scaled boundary finite element method is a novel semi‐analytical technique, whose versatility, accuracy and efficiency are not only equal to, but potentially better than the finite element method and the boundary element method for certain problems. This paper investigates the possibility of using higher‐order polynomial functions for the shape functions. Two techniques for generating the higher‐order shape functions are investigated. In the first, the spectral element approach is used with Lagrange interpolation functions. In the second, hierarchical polynomial shape functions are employed to add new degrees of freedom into the domain without changing the existing ones, as in the p‐version of the finite element method. To check the accuracy of the proposed procedures, a plane strain problem for which an exact solution is available is employed. A more complex example involving three scaled boundary subdomains is also addressed. The rates of convergence of these examples under p‐refinement are compared with the corresponding rates of convergence achieved when uniform h‐refinement is used, allowing direct comparison of the computational cost of the two approaches. The results show that it is advantageous to use higher‐order elements, and that higher rates of convergence can be obtained using p‐refinement instead of h‐refinement. Copyright © 2005 John Wiley & Sons, Ltd.     

Application of the X‐FEM to the fracture of piezoelectric materials

This paper presents an application of the extended finite element method (X‐FEM) to the analysis of fracture in piezoelectric materials. These materials are increasingly used in actuators and sensors. New applications can be found as constituents of smart composites for adaptive electromechanical structures. Under in service loading, phenomena of crack initiation and propagation may occur due to high electromechanical field concentrations. In the past few years, the X‐FEM has been applied mostly to model cracks in structural materials. The present paper focuses at first on the definition of new enrichment functions suitable for cracks in piezoelectric structures. At second, generalized domain integrals are used for the determination of crack tip parameters. The approach is based on specific asymptotic crack tip solutions, derived for piezoelectric materials. We present convergence results in the energy norm and for the stress intensity factors, in various settings. Copyright © 2008 John Wiley & Sons, Ltd.