Acoustic modelling by BEM–FEM coupling procedures taking into account explicit and implicit multi‐domain decomposition techniques

The numerical modelling of interacting acoustic media by boundary element method–finite element method (BEM–FEM) coupling procedures is discussed here, taking into account time‐domain approaches. In this study, the global model is divided into different sub‐domains and each sub‐domain is analysed independently (considering BEM or FEM discretizations): the interaction between the different sub‐domains of the global model is accomplished by interface procedures. Numerical formulations based on FEM explicit and implicit time‐marching schemes are discussed, resulting in direct and optimized iterative BEM–FEM coupling techniques. A multi‐level time‐step algorithm is considered in order to improve the flexibility, accuracy and stability (especially when conditionally stable time‐marching procedures are employed) of the coupled analysis. At the end of the paper, numerical examples are presented, illustrating the potentialities and robustness of the proposed methodologies. Copyright © 2008 John Wiley & Sons, Ltd.     

New development in extended finite element modeling of large elasto‐plastic deformations

This paper presents new achievements in the extended finite element modeling of large elasto‐plastic deformation in solid problems. The computational technique is presented based on the extended finite element method (X‐FEM) coupled with the Lagrangian formulation in order to model arbitrary interfaces in large deformations. In X‐FEM, the material interfaces are represented independently of element boundaries, and the process is accomplished by partitioning the domain with some triangular sub‐elements whose Gauss points are used for integration of the domain of elements. The large elasto‐plastic deformation formulation is employed within the X‐FEM framework to simulate the non‐linear behavior of materials. The interface between two bodies is modeled by using the X‐FEM technique and applying the Heaviside‐ and level‐set‐based enrichment functions. Finally, several numerical examples are analyzed, including arbitrary material interfaces, to demonstrate the efficiency of the X‐FEM technique in large plasticity deformations. Copyright © 2008 John Wiley & Sons, Ltd.     

Contact between 3D beams with rectangular cross‐sections

In this paper frictionless contact between 3D beams is analysed. The beam model is used in which large displacements but small strains are allowed. The element is derived on the basis of updated Lagrangian formulation using physical shape functions with shear effect included. An effective contact‐search algorithm, which is necessary to determine an active set for the contact contribution treatment, is elaborated. The contact element uses the same set of physical shape functions as the beam element. A consistent linearization of contact contribution is derived and expressed in suitable matrix form, easy to use in FEM approximation. Several numerical examples depict the efficiency of the presented approach. Copyright © 2002 John Wiley & Sons, Ltd.     

Crack face contact in X‐FEM using a segment‐to‐segment approach

In this study, we propose a segment‐to‐segment contact formulation (mortar‐based) that uses Lagrange's multipliers to establish the contact between crack faces when modeled with the extended finite element method (X‐FEM) in 2D problems. It is shown that, in general, inaccuracies arise when the contact is formulated following a point‐to‐point approach. This is due to the non‐linear character of the X‐FEM interpolation along the crack faces that leads to crack face interpenetration. However, the segment‐to‐segment approach optimizes the fulfilment of the contact constraints along the whole crack segment, and in practice the contact is modeled precisely. Convergence studies for mesh sequences have been performed, showing the advantages of the proposed methodology. Copyright © 2009 John Wiley & Sons, Ltd.     

A local multigrid X‐FEM strategy for 3‐D crack propagation

A multiscale method for 3‐D crack propagation simulation in large structures is proposed. The method is based on the extended finite element method (X‐FEM). The asymptotic behavior of the crack front is accurately modeled using enriched elements and no remeshing is required during crack propagation. However, the different scales involved in fracture mechanics problems can differ by several orders of magnitude and industrial meshes are usually not designed to account for small cracks. Enrichments are therefore useless if the crack is too small compared with the element size. To overcome this drawback, a project combining different numerical techniques was started. The first step was the implementation of a global multigrid algorithm within the X‐FEM framework and was presented in a previous paper (Eur. J. Comput. Mech. 2007; 16 :161–182). This work emphasized the high efficiency in cpu time but highlighted that mesh refinement is required on localized areas only (cracks, inclusions, steep gradient zones). This paper aims at linking the different scales by using a local multigrid approach. The coupling of this technique with the X‐FEM is described and computational aspects dealing with intergrid operators, optimal multiscale enrichment strategy and level sets are pointed out. Examples illustrating the accuracy and efficiency of the method are given. Copyright © 2008 John Wiley & Sons, Ltd.     

X‐FEM in isogeometric analysis for linear fracture mechanics

The extended finite element method (X‐FEM) has proven to be an accurate, robust method for solving problems in fracture mechanics. X‐FEM has typically been used with elements using linear basis functions, although some work has been performed using quadratics. In the current work, the X‐FEM formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics. In comparison with X‐FEM with conventional finite elements of equal degree, the NURBS‐based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom (DOF). Results for linear through quartic NURBS basis functions are presented for a multiplicity of one or a multiplicity equal the degree. Copyright © 2011 John Wiley & Sons, Ltd.     

On the modelling of complex 3D bulk metal forming processes via the pseudo‐concentrations technique. Application to the simulation of the Mannesmann piercing process

The modelling of complex 3D metal forming processes using the flow formulation, implemented via the pseudo‐concentrations technique, requires the development of robust computational strategies for dealing with the velocity and pseudo‐concentration boundary conditions in the zone where the blank–tools contact is developed. A new algorithm, designed to fulfil those requirements, is presented in this paper. The Mannesmann piercing process is a metal forming operation used in industry for manufacturing metal seamless pipes. The results of the Mannesmann process finite element simulation are particularly dependent on the accuracy and stability of the algorithm used to describe the contact boundary conditions between the forming tools and the blank. The validation of the finite element model is performed by comparing the numerical predictions obtained using the new algorithm with the results of industrial tests. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical simulation of granular materials by an improved discrete element method

In this paper, we present an improved discrete element method based on the non‐smooth contact dynamics and the bi‐potential concept. The energy dissipated during the collisions is taken into account by means of restitution coefficients. The interaction between particles is modelled by Coulomb unilateral contact law with dry friction which is typically non‐associated: during the contact, the sliding vector is not normal to the friction cone. The main feature of our algorithm is to overcome this difficulty by means of the bi‐potential theory. It leads to an easy implement predictor–corrector scheme involving just an orthogonal projection onto the friction cone. Moreover the convergence test is based on an error estimator in constitutive law using the corner stone inequality of the bipotential. Then we present numerical simulations which show the robustness of our algorithm and the various possibilities of the software ‘MULTICOR’ developed with this approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Implicit boundary method for analysis using uniform B‐spline basis and structured grid

Several analysis techniques such as extended finite element method (X‐FEM) have been developed recently, which use structured grid for the analysis. Implicit boundary method uses implicit equations of the boundary to apply boundary conditions in X‐FEM framework using structured grids. Solution structures for test and trial functions are constructed using implicit equations such that the boundary conditions are satisfied even if there are no nodes on the boundary. In this paper, this method is applied for analysis using uniform B‐spline basis defined over a structured grid. Solution structures that are C¹ or C² continuous throughout the analysis domain can be constructed using B‐spline basis functions. As a structured grid does not conform to the geometry of the analysis domain, the boundaries of the analysis domain are defined independently using equations of the boundary curves/surfaces. Compared with conforming mesh, it is easier to generate structured grids that overlap the geometry and the elements in the grid are regular shaped and undistorted. Numerical examples are presented to demonstrate the performance of these B‐spline elements. The results are compared with analytical solutions as well as with traditional finite element solutions. Convergence studies for several examples show that B‐spline elements provide accurate solutions with fewer elements and nodes compared with traditional FEM. They also provide continuous stress and strain in the analysis domain, thus eliminating the need for smoothing stress/strain results. Copyright © 2008 John Wiley & Sons, Ltd.     

A combined space–time extended finite element method

The Newmark method for the numerical integration of second order equations has been extensively used and studied along the past fifty years for structural dynamics and various fields of mechanical engineering. Easy implementation and nice properties of this method and its derivatives for linear problems are appreciated but the main drawback is the treatment of discontinuities. Zienkiewicz proposed an approach using finite element concept in time, which allows a new look at the Newmark method. The idea of this paper is to propose, thanks to this approach, the use of a time partition of the unity method denoted Time Extended Finite Element Method (TX‐FEM) for improved numerical simulations of time discontinuities. An enriched basis of shape functions in time is used to capture with a good accuracy the non‐polynomial part of the solution. This formulation allows a suitable form of the time‐stepping formulae to study stability and energy conservation. The case of an enrichment with the Heaviside function is developed and can be seen as an alternative approach to time discontinuous Galerkin method (T‐DGM), stability and accuracy properties of which can be derived from those of the TX‐FEM. Then Space and Time X‐FEM (STX‐FEM) are combined to obtain a unified space–time discretization. This combined STX‐FEM appears to be a suitable technique for space–time discontinuous problems like dynamic crack propagation or other applications involving moving discontinuities. Copyright © 2005 John Wiley & Sons, Ltd.     

A meshfree‐enriched finite element method for compressible and near‐incompressible elasticity

In this paper, a two‐dimensional displacement‐based meshfree‐enriched FEM (ME‐FEM) is presented for the linear analysis of compressible and near‐incompressible planar elasticity. The ME‐FEM element is established by injecting a first‐order convex meshfree approximation into a low‐order finite element with an additional node. The convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the Kronecker‐delta property on the element boundaries. The gradient matrix of ME‐FEM element satisfies the integration constraint for nodal integration and the resultant ME‐FEM formulation is shown to pass the constant stress test for the compressible media. The ME‐FEM interpolation is an element‐wise meshfree interpolation and is proven to be discrete divergence‐free in the incompressible limit. To prevent possible pressure oscillation in the near‐incompressible problems, an area‐weighted strain smoothing scheme incorporated with the divergence‐free ME‐FEM interpolation is introduced to provide the smoothing on strains and pressure. With this smoothed strain field, the discrete equations are derived based on a modified Hu–Washizu variational principle. Several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near‐incompressible problems. Copyright © 2012 John Wiley & Sons, Ltd.     

An energy‐conserving scheme for dynamic crack growth using the eXtended finite element method

This paper proposes a generalization of the eXtended finite element method (X‐FEM) to model dynamic fracture and time‐dependent problems from a more general point of view, and gives a proof of the stability of the numerical scheme in the linear case. First, we study the stability conditions of Newmark‐type schemes for problems with evolving discretizations. We prove that the proposed enrichment strategy satisfies these conditions and also ensures energy conservation. Using this approach, as the crack propagates, the enrichment can evolve with no occurrence of instability or uncontrolled energy transfer. Then, we present a technique based on Lagrangian conservation for the estimation of dynamic stress intensity factors for arbitrary 2D cracks. The results presented for several applications are accurate for stationary or moving cracks. Copyright © 2005 John Wiley & Sons, Ltd.     

Three‐dimensional finite element computations for frictional contact problems with non‐associated sliding rule

This paper presents an algorithm for solving anisotropic frictional contact problems where the sliding rule is non‐associated.The algorithm is based on a variational formulation of the complex interface model that combine the classical unilateral contact law and an anisotropic friction model with a non‐associated slip rule. Both the friction condition and the sliding potential are elliptical and have the same principal axes but with different semi‐axes ratio. The frictional contact law and its inverse are derived from a single non‐differentiable scalar‐valued function, called a bi‐potential. The convexity properties of the bi‐potential permit to associate stationary principles with initial/boundary value problems. With the present formulation, the time‐integration of the frictional contact law takes the form of a projection onto a convex set and only one predictor–corrector step addresses all cases (sticking, sliding, no‐contact). A solution algorithm is presented and tested on a simple example that shows the strong influence of the slip rule on the frictional behaviour. Copyright 2004 John Wiley & Sons, Ltd.     

A quasi‐static crack growth simulation based on the singular ES‐FEM

In the edge‐based smoothed finite element method (ES‐FEM), one needs only the assumed displacement values (not the derivatives) on the boundary of the edge‐based smoothing domains to compute the stiffness matrix of the system. Adopting this important feature, a five‐node crack‐tip element is employed in this paper to produce a proper stress singularity near the crack tip based on a basic mesh of linear triangular elements that can be generated automatically for problems with complicated geometries. The singular ES‐FEM is then formulated and used to simulate the crack propagation in various settings, using a largely coarse mesh with a few layers of fine mesh near the crack tip. The results demonstrate that the singular ES‐FEM is much more accurate than X‐FEM and the existing FEM. Moreover, the excellent agreement between numerical results and the reference observations shows that the singular ES‐FEM offers an efficient and high‐quality solution for crack propagation problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Arbitrary discontinuities in space–time finite elements by level sets and X‐FEM

An enriched finite element method with arbitrary discontinuities in space–time is presented. The discontinuities are treated by the extended finite element method (X‐FEM), which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper‐surface which is described by level sets. A space–time weak form for conservation laws is developed where the Rankine–Hugoniot jump conditions are natural conditions of the weak form. The method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non‐linear Burgers' equations. By capturing the discontinuity in time as well as space, results are improved over capturing the discontinuity in space alone and the method is remarkably accurate. Implications to standard semi‐discretization X‐FEM formulations are also discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Theory of Sliding‐Mode Triboelectric Nanogenerators

The triboelectric nanogenerator (TENG) is a powerful approach toward new energy technology, especially for portable electronics. A theoretical model for the sliding‐mode TENG is presented in this work. The finite element method was utilized to characterize the distributions of electric potential, electric field, and charges on the metal electrodes of the TENG. Based on the FEM calculation, the semi‐analytical results from the interpolation method and the analytical V‐Q‐x relationship are built to study the sliding‐mode TENG. The analytical V‐Q‐x equation is validated through comparison with the semi‐analytical results. Furthermore, based on the analytical V‐Q‐x equation, dynamic output performance of sliding‐mode TENG is calculated with arbitrary load resistance, and good agreement with experimental data is achieved. The theory presented here is a milestone work for in‐depth understanding of the working mechanism of the sliding‐mode TENG, and provides a theoretical basis for further enhancement of the sliding‐mode TENG for both energy scavenging and self‐powered sensor applications.     

Stabilized global–local X‐FEM for 3D non‐planar frictional crack using relevant meshes

A stabilized global–local quasi‐static contact algorithm for 3D non‐planar frictional crack is presented in the X‐FEM/level set framework. A three‐field weak formulation is considered and allows an independent discretization of the bulk and the crack interface. Then, a fine discretization of the interface can be defined according to the possible complex contact state along the crack faces independently from the mesh in the bulk. Furthermore, an efficient stabilized non‐linear LATIN solver dedicated to contact and friction is proposed. It allows solving in a unified framework frictionless and frictional contact at the crack interface with a symmetric formulation, no iterations on the local stage (unilateral contact law with/without friction), no calculation of any global tangent operator, and improved convergence rate. 2D and 3D patch tests are presented to illustrate the relevance of the proposed model and an actual 3D frictional crack problem under cyclic fretting loading is modeled. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient explicit time stepping for the eXtended Finite Element Method (X‐FEM)

This paper focuses on the introduction of a lumped mass matrix for enriched elements, which enables one to use a pure explicit formulation in X‐FEM applications. A proof of stability for the 1D and 2D cases is given. We show that if one uses this technique, the critical time step does not tend to zero as the support of the discontinuity reaches the boundaries of the elements. We also show that the X‐FEM element's critical time step is of the same order as that of the corresponding element without extended degrees of freedom. Copyright © 2006 John Wiley & Sons, Ltd.     

Polyarc discrete element for efficiently simulating arbitrarily shaped 2D particles

A new two‐dimensional discrete element type, termed the ‘polyarc’ element is presented in this paper. Compared to other discrete element types, the new element is capable of representing any two‐dimensional convex particle shape with arbitrary angularity and elongation using a small number of shape parameters. Contact resolution between polyarc elements, which is the most computation‐extensive task in DEM simulation only involves simple closed‐form solutions. Two undesirable contact scenarios common for polygon elements can be avoided by the polyarc element, so the contact resolution algorithm for polyarc elements is simpler than that for polygon elements. The extra flexibility in particle shape representation induces little or no additional computational cost. The key algorithmic aspects of the new element, including the particle shape representation scheme, the quick neighbor search algorithm, the contact resolution algorithm, and the contact law are presented. The recommended contact law for the polyarc model was formulated on the basis of an evaluation of various contact law schemes for polygon type discrete elements. The capability and efficiency of the new element type were demonstrated through an investigation of strength anisotropy of a virtual sand consisting of a random mix of angular and smooth elongated particles subjected to biaxial compression tests. Copyright © 2011 John Wiley & Sons, Ltd.