The nsBETI method: an extension of the FETI method to non‐symmetrical BEM‐FEM coupled problems

The finite element tearing and interconnecting (FETI) method is recognized as an effective domain decomposition tool to achieve scalability in the solution of partitioned second‐order elasticity problems. In the boundary element tearing and interconnecting (BETI) method, a direct extension of the FETI algorithm to the BEM, the symmetric Galerkin BEM formulation, is used to obtain symmetric system matrices, making possible to apply the same FETI conjugate gradient solver. In this work, we propose a new BETI variant labeled nsBETI that allows to couple substructures modeled with the FEM and/or non‐symmetrical BEM formulations. The method connects non‐matching BEM and FEM subdomains using localized Lagrange multipliers and solves the associated non‐symmetrical flexibility equations with a Bi‐CGstab iterative algorithm. Scalability issues of nsBETI in BEM–BEM and combined BEM–FEM coupled problems are also investigated. Copyright © 2012 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.     

Non‐linear analysis of thin‐walled members of open cross‐section

The paper presents a means of determining the non‐linear stiffness matrices from expressions for the first and second variation of the Total Potential of a thin‐walled open section finite element that lead to non‐linear stiffness equations. These non‐linear equations can be solved for moderate to large displacements. The variations of the Total Potential have been developed elsewhere by the authors, and their contribution to the various non‐linear matrices is stated herein. It is shown that the method of solution of the non‐linear stiffness matrices is problem dependent. The finite element procedure is used to study non‐linear torsion that illustrates torsional hardening, and the Newton–Raphson method is deployed for this study. However, it is shown that this solution strategy is unsuitable for the second example, namely that of the post‐buckling response of a cantilever, and a direct iteration method is described. The good agreement for both of these problems with the work of independent researchers validates the non‐linear finite element method of analysis. Copyright © 1999 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.     

Stress‐based finite element analysis of plane plasticity problems

A stress‐based model of the finite element method is evolved for two‐dimensional quasi‐static plasticity problems. The self‐equilibrating fields of stresses are constructed by means of the Airy stress function, which is approximated by three types of elements: the Bogner–Fox–Schmit rectangle, the Hsieh–Clough–Tocher triangle and its reduced variant. Traction boundary conditions are imposed by the use of the Lagrange multiplier method which gives the possibility of calculation of displacements for boundary points. The concept of multi‐point‐constraints elements is applied in order to facilitate the application of this technique. The iterative algorithm, analogous to the closest‐point‐projection method commonly used in the displacement‐based finite element model, is proposed for solving non‐linear equations for each load increment. Two numerical examples with stress‐ and displacement‐controlled load are considered. The results are compared with those obtained by the displacement model of FEM. Bounds for limit loads are obtained. Copyright © 1999 John Wiley & Sons, Ltd.     

Interface dynamic stiffness matrix approach for three‐dimensional transient multi‐region boundary element analysis

A novel substructuring method is developed for the coupling of boundary element and finite element subdomains in order to model three‐dimensional multi‐region elastodynamic problems in the time domain. The proposed procedure is based on the interface stiffness matrix approach for static multi‐region problems using variational principles together with the concept of Duhamel integrals. Unit impulses are applied at the boundary of each region in order to evaluate the impulse response matrices of the Duhamel (convolution) integrals. Although the method is not restricted to a special discretization technique, the regions are discretized using the boundary element method combined with the convolution quadrature method. This results in a time‐domain methodology with the advantages of performing computations in the Laplace domain, which produces very accurate and stable results as verified on test examples. In addition, the assembly of the boundary element regions and the coupling to finite elements are greatly simplified and more efficient. Finally, practical applications in the area of soil–structure interaction and tunneling problems are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

A combined FIC‐TDG finite element approach for the numerical solution of coupled advection–diffusion–reaction equations with application to a bioregulatory model for bone fracture healing

Numerical schemes for the approximative solution of advection–diffusion–reaction equations are often flawed because of spurious oscillations, caused by steep gradients or dominant advection or reaction. In addition, for strong coupled nonlinear processes, which may be described by a set of hyperbolic PDEs, established time stepping schemes lack either accuracy or stability to provide a reliable solution. In this contribution, an advanced numerical scheme for this class of problems is suggested by combining sophisticated stabilization techniques, namely the finite calculus (FIC‐FEM) scheme introduced by Oñate et al. with time‐discontinuous Galerkin (TDG) methods. Whereas the former one provides a stabilization technique for the numerical treatment of steep gradients for advection‐dominated problems, the latter ensures reliable solutions with regard to the temporal evolution. A brief theoretical outline on the superior behavior of both approaches will be presented and underlined with related computational tests. The performance of the suggested FIC‐TDG finite element approach will be discussed exemplarily on a bioregulatory model for bone fracture healing proposed by Geris et al., which consists of at least 12 coupled hyperbolic evolution equations. Copyright © 2012 John Wiley & Sons, Ltd.     

A coupling of multi‐zone curved Galerkin BEM with finite elements for independently modelled sub‐domains with non‐matching nodes in elasticity

When the different parts of a structure are modelled independently by BEM or FEM methods, it is sometimes necessary to put the parts together without remeshing of the nodes along the part interfaces. Frequently the nodes do not match along the interface. In this work, the symmetric Galerkin multi‐zone curved boundary element is a fully symmetric formulation and is the method used for the boundary element part. For BEM–FEM coupling it is then necessary to interpolate the tractions in‐between the non‐matching nodes for the FEM part. Finally, the coupling is achieved by transforming the finite element domains to equivalent boundary element domains in a block symmetric formulation. This system is then coupled with a boundary element domain with non‐matching nodes in‐between. Copyright © 2004 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.     

A spline wavelet finite‐element method in structural mechanics

The wavelet‐based methods are powerful to analyse the field problems with changes in gradients and singularities due to the excellent multi‐resolution properties of wavelet functions. Wavelet‐based finite elements are often constructed in the wavelet space where field displacements are expressed as a product of wavelet functions and wavelet coefficients. When a complex structural problem is analysed, the interface between different elements and boundary conditions cannot be easily treated as in the case of conventional finite‐element methods (FEMs). A new wavelet‐based FEM in structural mechanics is proposed in the paper by using the spline wavelets, in which the formulation is developed in a similar way of conventional displacement‐based FEM. The spline wavelet functions are used as the element displacement interpolation functions and the shape functions are expressed by wavelets. The detailed formulations of typical spline wavelet elements such as plane beam element, in‐plane triangular element, in‐plane rectangular element, tetrahedral solid element, and hexahedral solid element are derived. The numerical examples have illustrated that the proposed spline wavelet finite‐element formulation achieves a high numerical accuracy and fast convergence rate. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Non‐linear random response of laminated composite shallow shells using finite element modal method

This paper investigates the large‐amplitude multi‐mode random response of thin shallow shells with rectangular planform at elevated temperatures using a finite element non‐linear modal formulation. A thin laminated composite shallow shell element and the system equations of motion are developed. The system equations in structural node degrees‐of‐freedom (DOF) are transformed into modal co‐ordinates, and the non‐linear stiffness matrices are transformed into non‐linear modal stiffness matrices. The number of modal equations is much smaller than the number of equations in structural node DOF. A numerical integration is employed to determine the random response. Thermal buckling deflections are obtained to explain the intermittent snap‐through phenomenon. The natural frequencies of the infinitesimal vibration about the thermally buckled equilibrium positions (BEPs) are studied, and it is found that there is great difference between the frequencies about the primary (positive) and the secondary (negative) BEPs. All three types of motion: (i) linear random vibration about the primary BEP, (ii) intermittent snap‐through between the two BEPs, and (iii) non‐linear large‐amplitude random vibration over the two BEPs, can be predicted. Copyright © 2006 John Wiley & Sons, Ltd.     

Contact with friction in multi‐material arbitrary Lagrangian‐Eulerian formulations using X‐FEM

A contact method with friction for the multi‐dimensional Lagrangian step in multi‐material arbitrary Lagrangian–Eulerian (ALE) formulations is presented. In our previous research, the extended finite element method (X‐FEM) was used to create independent fields (i.e. velocity, strain rate, force, mass, etc.) for each material in the problem to model contact without friction. The research presented here includes the extension to friction and improvements to the accuracy and robustness of our previous study. The accelerations of the multi‐material nodes are obtained by coupling the material force and mass fields as a function of the prescribed contact; similarly, the velocities of the multi‐material nodes are recalculated using the conservation of momentum when the prescribed contact requires it. The coupling procedures impose the same nodal velocity on the coupled materials in the direction normal to their interface during the time step update. As a result, the overlap of materials is prevented and unwanted separation does not occur. Three different types of contacts are treated: perfectly bonded, frictionless slip, and slip with friction. Example impact problems are solved and the numerical solutions are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Bubble‐enhanced smoothed finite element formulation: a variational multi‐scale approach for volume‐constrained problems in two‐dimensional linear elasticity

This paper presents a bubble‐enhanced smoothed finite element formulation for the analysis of volume‐constrained problems in two‐dimensional linear elasticity. The new formulation is derived based on the variational multi‐scale approach in which unequal order displacement‐pressure pairs are used for the mixed finite element approximation and hierarchical bubble function is selected for the fine‐scale displacement approximation. An area‐weighted averaging scheme is employed for the two‐scale smoothed strain calculation under the framework of edge‐based smoothed FEM. The smoothed fine‐scale solution is shown to naturally contain the stress field jump of the smoothed coarse‐scale solution across the boundary of edge‐based smoothing domain and thus provides the possibility to stabilize the global solution for volume‐constrained problems. A global monolithic solution strategy is employed, and the fine‐scale solution is solved without the consideration of approximating the strong form of the fine‐scale equation. Several numerical examples are analyzed to demonstrate the accuracy of the present formulation. Copyright © 2014 John Wiley & Sons, Ltd.     

An efficient sparse matrix‐vector multiplication on CUDA‐enabled graphic processing units for finite element method simulations

Finite element method (FEM) is a well‐developed method to solve real‐world problems that can be modeled with differential equations. As the available computational power increases, complex and large‐size problems can be solved using FEM, which typically involves multiple degrees of freedom (DOF) per node, high order of elements, and an iterative solver requiring several sparse matrix‐vector multiplication operations. In this work, a new storage scheme is proposed for sparse matrices arising from FEM simulations with multiple DOF per node. A sparse matrix‐vector multiplication kernel and its variants using the proposed scheme are also given for CUDA‐enabled GPUs. The proposed scheme and the kernels rely on the mesh connectivity data from FEM discretization and the number of DOF per node. The proposed kernel performance was evaluated on seven test matrices for double‐precision floating point operations. The performance analysis showed that the proposed GPU kernel outperforms the ELLPACK (ELL) and CUSPARSE Hybrid (HYB) format GPU kernels by an average of 42% and 32%, respectively, on a Tesla K20c card. Copyright © 2016 John Wiley & Sons, Ltd.     

A domain decomposition method for a two‐phase transport model of polymer electrolyte fuel cell containing micro‐porous layer

In this paper, a nonlinear Dirichlet–Robin iteration‐by‐subdomain domain decomposition method is studied for a multidimensional, multiphysics, and multiphase model of polymer electrolyte fuel cell (PEFC) containing micro‐porous layer (MPL). Across the interface of gas diffusion layer and MPL in PEFC, it is well known that the capillary pressure is continuous, whereas liquid saturation is discontinuous, by which the liquid‐water removal in the porous electrodes can be significantly enhanced. We design a type of non‐overlapping domain decomposition method to deal with water transport in such multi‐layer diffusion media, where Kirchhoff transformation and its inverse techniques are employed to conquer the discontinuous and degenerate water diffusivity in the coexisting single‐phase and two‐phase regions. In addition, the conservation equations of mass, momentum, charge, and hydrogen and oxygen transport are also solved by the combined finite element–upwind finite volume method (FEM/FVM) to overcome the dominated convection effect in gas channels. Numerical simulations demonstrate that the presented techniques are effective in obtaining a fast and convergent nonlinear iteration for such a 3D PEFC model within around 50 steps, in contrast with the oscillatory and nonconvergent iteration conducted by standard FEM/FVM. A series of numerical convergence tests are also carried out to verify the efficiency and accuracy of the present numerical techniques. Copyright © 2012 John Wiley & Sons, Ltd.     

A particle‐in‐cell solution to the silo discharging problem

The problem of flow of a granular material during the process of discharging a silo is considered in the present paper. The mechanical behaviour of the material is described by the use of the model of the elastic–plastic solid with the Drucker–Prager yield condition and the non‐associative flow rule. The phenomenon of friction between the stored material and the silo walls is taken into account—the Coulomb model of friction is used in the analysis. The problem is analysed by means of the particle‐in‐cell method—a variant of the finite element method which enables to solve the pertinent equations of motion on an arbitrary computational mesh and trace state variables at points of the body chosen independently of the mesh. The method can be regarded as an arbitrary Lagrangian–Eulerian formulation of the finite element method, and overcomes the main drawback of the updated Lagrangian formulation of FEM related to mesh distortion. The entire process of discharging a silo can be analysed by this approach. The dynamic problem is solved by the use of the explicit time‐integration scheme. Several numerical examples are included. The plane strain and axisymmetric problems are solved for silos with flat bottoms and conical hoppers. Some results are compared with experimental ones. Copyright © 1999 John Wiley & Sons, Ltd.     

A coupled mechanical‐charge/dipole molecular dynamics finite element method,with multi‐scale applications to the design of graphene nano‐devices

A new Molecular Dynamics Finite Element Method (MDFEM) with a coupled mechanical‐charge/dipole formulation is proposed. The equilibrium equations of Molecular Dynamics (MD) are embedded exactly within the computationally more favourable Finite Element Method (FEM). This MDFEM can readily implement any force field because the constitutive relations are explicitly uncoupled from the corresponding geometric element topologies. This formal uncoupling allows to differentiate between chemical‐constitutive, geometric and mixed‐mode instabilities. Different force fields, including bond‐order reactive and polarisable fluctuating charge–dipole potentials, are implemented exactly in both explicit and implicit dynamic commercial finite element code. The implicit formulation allows for larger length and time scales and more varied eigenvalue‐based solution strategies. The proposed multi‐physics and multi‐scale compatible MDFEM is shown to be equivalent to MD, as demonstrated by examples of fracture in carbon nanotubes (CNT), and electric charge distribution in graphene, but at a considerably reduced computational cost. The proposed MDFEM is shown to scale linearly, with concurrent continuum FEM multi‐scale couplings allowing for further computational savings. Moreover, novel conformational analyses of pillared graphene structures (PGS) are produced. The proposed model finds potential applications in the parametric topology and numerical design studies of nano‐structures for desired electro‐mechanical properties (e.g. stiffness, toughness and electric field induced vibrational/electron‐emission properties). Copyright © 2014 John Wiley & Sons, Ltd.     

A face‐based smoothed finite element method (FS‐FEM) for 3D linear and geometrically non‐linear solid mechanics problems using 4‐node tetrahedral elements

This paper presents a novel face‐based smoothed finite element method (FS‐FEM) to improve the accuracy of the finite element method (FEM) for three‐dimensional (3D) problems. The FS‐FEM uses 4‐node tetrahedral elements that can be generated automatically for complicated domains. In the FS‐FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements. The results demonstrated that the FS‐FEM is significantly more accurate than the FEM using tetrahedral elements for both linear and geometrically non‐linear solid mechanics problems. In addition, a novel domain‐based selective scheme is proposed leading to a combined FS/NS‐FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The implementation of the FS‐FEM is straightforward and no penalty parameters or additional degrees of freedom are used. The computational efficiency of the FS‐FEM is found better than that of the FEM. Copyright © 2008 John Wiley & Sons, Ltd.     

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

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