Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations

This paper presents the combination of new mesh-free radial basis function network (RBFN) methods and domain decomposition (DD) technique for approximating functions and solving Poisson's equations. The RBFN method allows numerical approximation of functions and solution of partial differential equations (PDEs) without the need for a traditional ‘finite element’-type (FE) mesh while the combined RBFN–DD approach facilitates coarse-grained parallelisation of large problems. Effect of RBFN parameters on the quality of approximation of function and its derivatives is investigated and compared with the case of single domain. In solving Poisson's equations, an iterative procedure is employed to update unknown boundary conditions at interfaces. At each iteration, the interface boundary conditions are first estimated by using boundary integral equations (BIEs) and subdomain problems are then solved by using the RBFN method. Volume integrals in standard integral equation representation (IE), which usually require volume discretisation, are completely eliminated in order to preserve the mesh-free nature of RBFN methods. The numerical examples show that RBFN methods in conjunction with DD technique achieve not only a reduction of memory requirement but also a high accuracy of the solution.     

Adaptive modeling of damage growth using a coupled FEM/BEM approach

We propose a coupled boundary element method (BEM) and a finite element method (FEM) for modelling localized damage growth in structures. BEM offers the flexibility of modelling large domains efficiently, while the non‐linear damage growth is accurately accounted by a local FEM mesh. An integral‐type nonlocal continuum damage mechanics with adapting FEM mesh is used to model multiple damage zones and follow their propagation in the structure. Strong form coupling, BEM hosted, is achieved using Lagrange multipliers. Because the non‐linearity is isolated in the FEM part of the system of equations, the system size is reduced using Schur complement approach, then the solution is obtained by a monolithic Newton method that is used to solve both domains simultaneously. The coupled BEM/FEM approach is verified by a set of convergence studies, where the reference solution is obtained by a fine FEM. In addition, the method is applied to multiple fractures growth benchmark problems and shows good agreement with the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Shear deformation effect in second-order analysis of frames subjected to variable axial loading

In this paper a boundary element method is developed for the second-order analysis of frames consisting of beams of arbitrary simply or multiply connected constant cross section, taking into account shear deformation effect. Each beam is subjected to an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Three boundary value problems are formulated with respect to the beam deflection, the axial displacement and to a stress function and solved employing a BEM approach. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress function using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The influence of both the shear deformation effect and the variableness of the axial loading are remarkable.     

Shear deformation effect in nonlinear analysis of spatial composite beams subjected to variable axial loading by bem

Summary In this paper a boundary element method is developed for the nonlinear analysis of composite beams of arbitrary doubly symmetric constant cross section, taking into account the shear deformation effect. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson's ratio and are firmly bonded together. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency and the range of applications of the developed method. The influence of both the shear deformation effect and the variableness of the axial loading are remarkable.     

Flexural - Torsional Nonlinear Analysis of Timoshenko Beam-Column of Arbitrary Cross Section by BEM

In this paper a boundary element method is developed for the nonlinear flexural - torsional analysis of Timoshenko beam-columns of arbitrary simply or multiply connected constant cross section, undergoing moderate large deflections under general boundary conditions. The beam-column is subjected to the combined action of an arbitrarily distributed or concentrated axial and transverse loading as well as to bending and twisting moments. To account for shear deformations, the concept of shear deformation coefficients is used. Seven boundary value problems are formulated with respect to the transverse displacements, to the axial displacement, to the angle of twist (which is assumed to be small), to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements as well as the angle of twist are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The influence of both the shear deformation effect and the variableness of the axial loading are remarkable.     

A two‐scale generalized finite element approach for modeling localized thermoplasticity

Predicting localized, nonlinear, thermoplastic behavior and residual stresses and deformations in structures subjected to intense heating is a prevalent challenge in a range of modern engineering applications. The authors present a generalized finite element method targeted at this class of problems, involving the solution of intrinsically parallelizable local boundary value problems to capture localized, time‐dependent thermo‐elasto‐plastic behavior, which is embedded in the coarse, structural‐scale approximation via enrichment functions. The method accommodates approximation spaces that evolve in between time or load steps while maintaining a fixed global mesh, which avoids the need to map solutions and state variables on changing meshes typical of traditional adaptive approaches. Representative three‐dimensional examples exhibiting localized, transient, nonlinear thermal and thermomechanical effects are presented to demonstrate the advantages of the method with respect to available approaches, especially in terms of its flexibility and potential for realistic future applications in this area. Parallelism of the approach is also discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

An efficient multi‐layer planar 3D fracture growth algorithm using a fixed mesh approach

We present a planar three‐dimensional (3D) fracture growth simulator, based on a displacement discontinuity (DD) method for multi‐layer elasticity problems. The method uses a fixed mesh approach, with rectangular panel elements to represent the planar fracture surface. Special fracture tip logic is included that allows a tip element to be partially fractured in the tip region. The fracture perimeter is modelled in a piece‐wise linear manner. The algorithm can model any number of interacting fractures that are restricted to lie on a single planar surface, located orthogonal to any number of parallel layers. The multiple layers are treated using a Fourier transform (FT) approach that provides a numerical Green's function for the DD scheme. The layers are assumed to be fully bonded together. Any fracture growth rule can be postulated for the algorithm. We demonstrate this approach on a number of test problems to verify its accuracy and efficiency, before showing some more general results. Copyright © 2001 John Wiley & Sons, Ltd.     

A domain decomposition method for discontinuous Galerkin discretizations of Helmholtz problems with plane waves and Lagrange multipliers

A nonoverlapping domain decomposition (DD) method is proposed for the iterative solution of systems of equations arising from the discretization of Helmholtz problems by the discontinuous enrichment method. This discretization method is a discontinuous Galerkin finite element method with plane wave basis functions for approximating locally the solution and dual Lagrange multipliers for weakly enforcing its continuity over the element interfaces. The primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface Lagrange multipliers only. As in the FETI‐H and FETI‐DPH DD methods for continuous Galerkin discretizations, this system of Lagrange multipliers is iteratively solved by a Krylov method equipped with both a local preconditioner based on subdomain data, and a global one using a coarse space. Numerical experiments performed for two‐ and three‐dimensional acoustic scattering problems suggest that the proposed DD‐based iterative solver is scalable with respect to both the size of the global problem and the number of subdomains. Copyright © 2009 John Wiley & Sons, Ltd.     

Uniform asymptotic approximations for accurate modeling of cracks in layered elastic media*

We present uniform asymptotic solutions (UAS) for displacement discontinuities (DD) that lie within the middle layer of a three layer elastic medium. The DDs are assumed to be normal to the two parallel interfaces between the leastic media, and solutions will be presented for both 2D and 3D elastic media. Using the Fourier transform (FT) method we construct the leading term in the asymptotic expansion for the spectral coefficient functions for a DD in a three layer medium. Although a closed form solution will require an infinite series solution, we demonstrate how this UAS can be used to construct highly efficient and accurate solutions even in the case in which the DD actually touches the interface. We present an explicit UAS for elements in which the DD fields are assumed to be piecewise constant throughout a line segment in 2D and a rectangular element in 3D. We demonstrate the usefulness of this UAS by providing a number of examples in which the UAS is used to solve problems in which cracks just touch or cross an interface. The accuracy and efficiency of the algorithm is demonstrated and compared with other numerical methods such as the finite element method and the boudary integral method.     

Analysis of plates stiffened by parallel beams

In this paper a general solution for the analysis of plates stiffened by parallel beams subjected to an arbitrary loading is presented. According to the proposed model, the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. The aforementioned integrated tractions result in the loading of the beams as well as the additional loading of the plate. Their distribution is established by applying continuity conditions in all directions at the interfaces. The analysis of both the plate and the beams is accomplished on their deformed shape taking into account second‐order effects. Six boundary value problems with respect to the plate transverse deflection, to the plate inplane displacement components, to the beam transverse deflections, to the beam axial deformation and to the beam non‐uniform angle of twist are formulated and solved using the analog equation method (AEM), a boundary element method (BEM)‐based method employing a boundary integral equation approach. The solution of the aforementioned plate and beam problems, which are non‐linearly coupled, is achieved using iterative numerical methods. The adopted model describes better the actual response of the plate beams system and permits the evaluation of the shear forces at the interfaces in both directions, the knowledge of which is very important in the design of prefabricated ribbed plates. The evaluated lateral deflections of the plate–beams system are found to exhibit considerable discrepancy from those of other models, which neglect inplane and axial forces and deformations. Copyright © 2006 John Wiley & Sons, Ltd.     

Computational multiscale modelling of heterogeneous material layers

A computational homogenization procedure for a material layer that possesses an underlying heterogeneous microstructure is introduced within the framework of finite deformations. The macroscopic material properties of the material layer are obtained from multiscale considerations. At the macro level, the layer is resolved as a cohesive interface situated within a continuum, and its underlying microstructure along the interface is treated as a continuous representative volume element of given height. The scales are linked via homogenization with customized hybrid boundary conditions on this representative volume element, which account for the deformation modes along the interface. A nested numerical solution scheme is adopted to link the macro and micro scales. Numerical examples successfully display the capability of the proposed approach to solve macroscopic boundary value problems with an evaluation of the constitutive properties of the material layer based on its micro-constitution.     

A continuum sensitivity method for finite thermo‐inelastic deformations with applications to the design of hot forming processes

A computational framework is presented to evaluate the shape as well as non‐shape (parameter) sensitivity of finite thermo‐inelastic deformations using the continuum sensitivity method (CSM). Weak sensitivity equations are developed for the large thermo‐mechanical deformation of hyperelastic thermo‐viscoplastic materials that are consistent with the kinematic, constitutive, contact and thermal analyses used in the solution of the direct deformation problem. The sensitivities are defined in a rigorous sense and the sensitivity analysis is performed in an infinite‐dimensional continuum framework. The effects of perturbation in the preform, die surface, or other process parameters are carefully considered in the CSM development for the computation of the die temperature sensitivity fields. The direct deformation and sensitivity deformation problems are solved using the finite element method. The results of the continuum sensitivity analysis are validated extensively by a comparison with those obtained by finite difference approximations (i.e. using the solution of a deformation problem with perturbed design variables). The effectiveness of the method is demonstrated with a number of applications in the design optimization of metal forming processes. Copyright © 2002 John Wiley & Sons, Ltd.     

A fast implementation of the FETI‐DP method: FETI‐DP‐RBS‐LNA and applications on large scale problems with localized non‐linearities

As parallel and distributed computing gradually becomes the computing standard for large scale problems, the domain decomposition method (DD) has received growing attention since it provides a natural basis for splitting a large problem into many small problems, which can be submitted to individual computing nodes and processed in a parallel fashion. This approach not only provides a method to solve large scale problems that are not solvable on a single computer by using direct sparse solvers but also gives a flexible solution to deal with large scale problems with localized non‐linearities. When some parts of the structure are modified, only the corresponding subdomains and the interface equation that connects all the subdomains need to be recomputed. In this paper, the dual–primal finite element tearing and interconnecting method (FETI‐DP) is carefully investigated, and a reduced back‐substitution (RBS) algorithm is proposed to accelerate the time‐consuming preconditioned conjugate gradient (PCG) iterations involved in the interface problems. Linear–non‐linear analysis (LNA) is also adopted for large scale problems with localized non‐linearities based on subdomain linear–non‐linear identification criteria. This combined approach is named as the FETI‐DP‐RBS‐LNA algorithm and demonstrated on the mechanical analyses of a welding problem. Serial CPU costs of this algorithm are measured at each solution stage and compared with that from the IBM Watson direct sparse solver and the FETI‐DP method. The results demonstrate the effectiveness of the proposed computational approach for simulating welding problems, which is representative of a large class of three‐dimensional large scale problems with localized non‐linearities. Copyright © 2005 John Wiley & Sons, Ltd.     

Nonlinear thermal analysis with a boundary element zone condensation technique

This paper presents a substantially more economical technique for the boundary element analysis (BEA) of a large class of nonlinear heat transfer problems including those with temperature dependent conductivity, temperature dependent convection coefficients, and radiation boundary conditions. The technique involves an exact static condensation of boundary element zones in a multi-zone boundary element model. The condensed boundary element zone contributions to be overall sparse blocked boundary element system matrices are formed once in the first step of the iterative nonlinear solution process and subsequently reused as the nonlinear parts of the overall problem are evolved to a convergent solution. Through a series of example problems it is demonstrated that the zone condensation technique facilitates the use of highly convergent iterative strategies for the solution of the nonlinear heat transfer problem involving modification and subsequent factorization of the overall boundary element system left had side matrix. For heat transfer problems with localized nonlinear effects, the condensation technique is shown to allow for the solution of nonlinear problems in less than half the CPU time required by methods that do not employ condensation.     

Patch coupling in isogeometric analysis of solids in boundary representation using a mortar approach

This contribution is concerned with a coupling approach for nonconforming NURBS patches in the framework of an isogeometric formulation for solids in boundary representation. The boundary representation modeling technique in CAD is the starting point of this approach. We parameterize the solid according to the scaled boundary finite element method and employ NURBS basis functions for the approximation of the solution. Therefore, solid surfaces consist of several sections, which can be regarded as patches and discretized independently. The main objective of this study is to derive an approach for the connection of independent sections in order to allow for local refinement and thus an accurate and efficient discretization of the computational domain. Nonconforming sections are coupled with a mortar approach within a master-slave framework. The coupling of adjacent sections ensures the equality of mutual deformations along the interface in a weak sense and is enforced by constraining the NURBS basis functions on the interface. We apply this approach to nonlinear problems in two dimensions and compare the results with conforming discretizations.     

Shear deformation effect in flexural–torsional vibrations of beams by BEM

In this paper, a boundary element method is developed for the general flexural–torsional vibration problem of Timoshenko beams of arbitrarily shaped cross section taking into account the effects of warping stiffness, warping and rotary inertia and shear deformation. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy.     

Coupled thermo-mechanical analysis of metal-forming processes through a combined finite element–boundary element approach

This paper presents a new approach for obtaining the distribution of temperature in the dies during thermo-mechanical numerical analysis of metal forming problems. The proposed approach is based on a solution resulting from the combination of the finite element method with the boundary element method. The finite element method is used to perform the numerical modelling of the thermo-mechanical deformation of the workpiece, taking into account the geometrical and material non-linearities as well as the influence of the temperature distribution on the mechanical behaviour of the material. The boundary element method is applied for computing the distribution of temperatures in the dies. The combination of the two numerical methods is made using the finite element solution of the heat flow exchanged across the die–workpiece interface to define the boundary conditions to be applied on the thermal analysis of the dies. A numerical example of compression under plane-strain conditions is included to show the applicability of the proposed approach. © 1998 John Wiley & Sons, Ltd.     

A finite element-boundary element treatment for the analysis of elastohydrodynamic lubrication problems

A combined finite element evaluation of hydrodynamic pressure and boundary element calculation of film thickness is presented. The viscosity and the density of the oil are assumed to vary with pressure, however the isothermal condition is assumed to prevail. The technique is based on an iterative procedure by assuming an initial hydrodynamic pressure. The iteration cycle will then be followed by the calculation of the film thickness and hydrodynamic pressure to arrive at a converged solution. The bearings have been treated realistically as finite domain bodies and their deformations are evaluated by boundary element method. The accuracy of the technique is illustrated in elastohydrodynamic lubrication (EHL) of inclined slider and line contact problems.     

Efficient boundary element analysis of sharp notched plates

The present paper further develops the boundary element singularity subtraction technique, to provide an efficient and accurate method of analysing the general mixed-mode deformation of two-dimensional linear elastic structures containing sharp notches. The elastic field around sharp notches is singular. Because of the convergence difficulties that arise in numerical modelling of elastostatic problems with singular fields, these singularities are subtracted out of the original elastic field, using the first term of the Williams series expansion. This regularization procedure introduces the stress intensity factors as additional unknowns in the problem; hence extra conditions are required to obtain a solution. Extra conditions are defined such that the local solution in the neighbourhood of the notch tip is identical to the Williams solution; the procedure can take into account any number of terms of the series expansion. The standard boundary element method is modified to handle additional unknowns and extra boundary conditions. Analysis of plates with symmetry boundary conditions is shown to be straightforward, with the modified boundary element method. In the case of non-symmetrical plates, the singular tip-tractions are not primary boundary element unknowns. The boundary element method must be further modified to introduce the boundary integral stress equations of an internal point, approaching the notch-tip, as primary unknowns in the formulation. The accuracy and efficiency of the method is demonstrated with some benchmark tests of mixed-mode problems. New results are presented for the mixed-mode analysis of a non-symmetrical configuration of a single edge notched plate.     

Finite deformation plasticity and viscoplasticity laws exhibiting nonlinear hardening rules

  We consider two finite deformation plasticity models, which differ mainly in the evolution equation governing the response of kinematic hardening. Both models reduce to the same constitutive law in the case of small deformations. The aim of the paper is to discuss these models by calculating the predicted responses for some representative loading conditions. The numerical calculations needed are performed by using an efficient time-integration algorithm which has been developed with a view to implementation in the ABAQUS finite element code. Generally, there are some differences between the predicted responses and in particular between the second-order effects predicted by the two models. For some simple deformation processes, e.g. simple shear and simple torsion, the differences concerning second-order effects exhibit some kind of regularities, which are independent of material parameters. Also, even if boundary value problems are considered where global deformations are small, significant differences can exist between the predicted model responses according to the finite deformation and the limiting small deformation theory.