Parameter identification for coupled finite element–boundary element models

To solve problems involving semi-infinite domains, one efficient approach is to use finite elements (FE) to model the regions where detailed information about materials with complicated properties is needed and to use boundary elements (BE) to simulate the semi-infinite parts. In this paper, a parameter identification algorithm is developed for coupled FE–BE models in detail. The algorithm is designed to identify all the material parameters in the FE domain and the BE domain simultaneously. Its validity is illustrated using two examples. The distribution of the observational points is also briefly tested and discussed. The numerical results reveal that this is a stable and fast-converging algorithm.     

3-D boundary element–finite element method for the dynamic analysis of piled buildings

A boundary element–finite element model is presented for the three-dimensional dynamic analysis of piled buildings in the frequency domain. Piles are modelled as compressible Euler–Bernoulli beams founded on a linear, isotropic, viscoelastic, zoned-homogeneous, unbounded layered soil, while multi-storey buildings are assumed to be comprised of vertical compressible piers and rigid slabs. Soil–foundation–structure interaction is rigorously taken into account with an affordable number of degrees of freedom. The code allows the direct analysis of multiple piled buildings, so that the influence of other constructions can be taken into account in the analysis of a certain element. The formulation is outlined before presenting validation results and an application example.     

Modelling multiple cohesive crack propagation using a finite element–scaled boundary finite element coupled method

This paper presents an extension of the recently-developed finite element–scaled boundary finite element (FEM–SBFEM) coupled method to model multiple crack propagation in concrete. The concrete bulk and fracture process zones are modelled using SBFEM and nonlinear cohesive interface finite elements (CIEs), respectively. The CIEs are automatically inserted into the SBFEM mesh as the cracks propagate. The algorithm previously devised for single crack propagation is augmented to model problems with multiple cracks and to allow cracks to initiate in an un-cracked SBFEM mesh. It also addresses crack propagation from one subdomain into another, as a result of partitioning a coarse SBFEM mesh, required for some mixed–mode problems. Each crack in the SBFEM mesh propagates when the sign of the Mode-I stress intensity factor at the crack tip turns positive from negative. Its propagation angle is determined using linear elastic fracture mechanics criteria. Three concrete beams involving multiple crack propagation are modelled. The predicted crack propagation patterns and load–displacement curves are in good agreement with data reported in literature.     

Efficient parallel algorithms for elastic–plastic finite element analysis

This paper presents our new development of parallel finite element algorithms for elastic–plastic problems. The proposed method is based on dividing the original structure under consideration into a number of substructures which are treated as isolated finite element models via the interface conditions. Throughout the analysis, each processor stores only the information relevant to its substructure and generates the local stiffness matrix. A parallel substructure oriented preconditioned conjugate gradient method, which is combined with MR smoothing and diagonal storage scheme are employed to solve linear systems of equations. After having obtained the displacements of the problem under consideration, a substepping scheme is used to integrate elastic–plastic stress–strain relations. The procedure outlined controls the error of the computed stress by choosing each substep size automatically according to a prescribed tolerance. The combination of these algorithms shows a good speedup when increasing the number of processors and the effective solution of 3D elastic–plastic problems whose size is much too large for a single workstation becomes possible.     

A 3-D six-noded finite element containing a λ singularity

A three-dimensional six-noded prism (wedge) finite element containing a singularity of order is developed. The interpolation functions of the displacement allow for variations proportional to the power of the distance from the crack front along the crack surface and to the distance in the perpendicular direction. The element is compatible with standard wedge (6-noded) and brick (8-noded) isoparametric elements. Three problems are studied to examine the proposed element, namely, a penny-shaped crack in a cylinder, an edge crack in an homogeneous material, and a crack perpendicular to a bimaterial interface.     

Four-node mixed Hu–Washizu shell element with drilling rotation

In this paper, enhanced four-node shell elements with six DOFs/node based on the Hu–Washizu (HW) functional are developed for Green strain. The drilling rotation is included through the drilling rotation constraint equation. The key features of the approach are as follows.

1. The shell HW functional is derived from the shell potential energy functional, which is an alternative to the derivation from the three-dimensional HW functional. This method is more versatile as it enables the derivation of the so-called partial HW functionals, with different treatment of the bending/twisting part and the transverse shear part of strain energy.
2. For the membrane part of HW shell elements, a seven-parameter stress, a nine-parameter strain and a two-parameter enhanced assumed displacement gradient enhancement are selected as optimal. The assumed representations of stress and strain are defined in skew coordinates in the natural basis at the element's center. This improves accuracy and has positive theoretical consequences.
3. The drilling rotation constraint equation is treated by the perturbed Lagrange method. The faulty term resulting from the equal-order approximations of displacements and the drilling rotation is eliminated, and one spurious mode is stabilized using the gamma method. The proposed formulation is insensitive to the element's distortions and yields a large radius of convergence in the examples involving in-plane bending.

The performance of 4 four-node shell HW elements, having different bending/twisting and transverse shear parts, is analyzed on several numerical examples. Such aspects are considered as: accuracy, radius of convergence, required number of iterations of the Newton method or the arc-length method and time of computations. The element with 29 parameters (HW29) is selected as the best performer. Copyright © 2011 John Wiley & Sons, Ltd.     

Elasto‐plastic deformation of upsetting by boundary element method

Abstract

This paper presents the utilization of boundary element method (BEM) to analyze the elasto‐plastic deformation of upsetting problems. Method of successive elastic solutions is used in the nonlinear analysis; both the linear strain hardening and the power law relation are used as constitutive equations of the material. For the later model the slope of strain hardening at each step is modified to a more correct prediction to make the deformation step larger and to obtain better convergence. The result may verify the stress‐strain curve as it does, and verify the similar pattern of the plastic zone propagation as Roll's result by finite element method. It is shown that various frictional conditions and width‐height ratios of the workpiece also influence the propagation behavior of plastic zones.     

The combined finite–discrete element method for structural failure and collapse

A combined finite–discrete element model for failure and collapse of structural systems comprising of reinforced concrete beam or column type structural members has been developed and implemented into a combined finite–discrete element code. The results obtained using the proposed model compare well with analytical and experimental results. In addition the rotational capacities obtained are in good agreement with published experimental results.     

Adaptive meshless local maximum-entropy finite element method for convection–diffusion problems

In this paper, a meshless local maximum-entropy finite element method (LME-FEM) is proposed to solve 1D Poisson equation and steady state convection–diffusion problems at various Peclet numbers in both 1D and 2D. By using local maximum-entropy (LME) approximation scheme to construct the element shape functions in the formulation of finite element method (FEM), additional nodes can be introduced within element without any mesh refinement to increase the accuracy of numerical approximation of unknown function, which procedure is similar to conventional p-refinement but without increasing the element connectivity to avoid the high conditioning matrix. The resulted LME-FEM preserves several significant characteristics of conventional FEM such as Kronecker-delta property on element vertices, partition of unity of shape function and exact reproduction of constant and linear functions. Furthermore, according to the essential properties of LME approximation scheme, nodes can be introduced in an arbitrary way and the $C^0$ continuity of the shape function along element edge is kept at the same time. No transition element is needed to connect elements of different orders. The property of arbitrary local refinement makes LME-FEM be a numerical method that can adaptively solve the numerical solutions of various problems where troublesome local mesh refinement is in general necessary to obtain reasonable solutions. Several numerical examples with dramatically varying solutions are presented to test the capability of the current method. The numerical results show that LME-FEM can obtain much better and stable solutions than conventional FEM with linear element.     

3D mass-redistributed finite element method in structural–acoustic interaction problems

A 2D mass-redistributed finite element method (MR-FEM) for pure acoustic problems was recently proposed to reduce the dispersion error. In this paper, the 3D MR-FEM is further developed to solve more complicated structural–acoustic interaction problems. The smoothed Galerkin weak form is adopted to formulate the discretized equations for the structure, and MR-FEM is applied in acoustic domain. The global equations of structural–acoustic interaction problems are then established by coupling the MR-FEM for the acoustic domain and the edge-based smoothed finite element method for the structure. The perfect balance between the mass matrix and stiffness matrix is able to improve the accuracy of the acoustic domain significantly. The gradient smoothing technique used in the structural domain can provide a proper softening effect to the “overly-stiff” FEM model. A number of numerical examples have demonstrated the effectiveness of the mass-redistributed method with smoothed strain.     

“Causal” shape functions in the time domain boundary element method

Since failing to respect the causality condition has been identified as one of the main sources of inaccuracies in the time domain boundary element method for elastodynamics and scalar wave propagation problems, in this contribution new shape functions are investigated, which permit a more accurate simulation of the continuous propagation of wave fronts. The performance of these shape functions in 2D scalar wave propagation problems is tested both for the potential (displacement) and for the time gradient (velocity) equations. Analytical time integrations are developed and numerical results are presented.     

Baseline rolling resistance for tires’ on-road fuel efficiency using finite element modeling

Calculation of truck tires rolling resistance, using the finite element method and considering variables such as incompressible visco-hyperelastic rubber materials, accurate tire geometry and steady temperature distribution, is presented. The model was validated using experimentally measured contact area and contact stresses. Rolling resistance was calculated for three values of axle load, tire inflation pressure, temperature and speed. In addition, regression analysis was used to propose a mathematical expression for predicting rolling resistance as a function of the considered variables. Finally, the contribution of tire’s rubber components to the internal energy was quantified, and it was found that sidewall and subtread were the most relevant. The results of this study will help differentiate the contribution of pavement parameters, such as mean profile depth and international roughness index, to fuel efficiency.     

The dual boundary element method:Ĵ-integral for dynamic stress intensity factors

The application of the dual boundary element method and the path independent-integral for the evaluation of dynamic stress intensity factors of stationary cracks in a linear elastic material is presented. The distinct set of boundary equations of elastodynamics is obtained by using the dual boundary element method and the dual reciprocity approach. Numerical implementation of the path-independent-integral and the decomposition technique is presented. The method is applied for several cracked structures and the results are compared with solutions obtained by using other methods.On leave from Silesian Technical University of Gliwice, Poland.     

Finite element modelling of rubber-like materials — a comparison between simulation and experiment

In this work finite element simulations are used based on the micro structure of polymers in order to transfer the information of the micro level to the macro level. The microscopic structure of polymers is characterized by a three-dimensional network consisting of randomly oriented chain-like macromolecules linked together at certain points. Different techniques are used to simulate the rubber-like material behaviour of such networks. These techniques range from molecular dynamics to the finite element method.The proposed approach is based on a so-called unit cell. This unit cell consists of one tetrahedral element and six truss elements. To each edge of the tetrahedron one truss element is attached which models the force-stretch behaviour of a bundle of polymer chains. The proposed method provides the possibility to observe how changes at the microscopic level influence the macroscopic material behaviour. Such observations were carried out in [1]. The main focus of this work is the validation of the proposed approach. Therefore the model is compared to different experimental data and other statistically-based network models describing rubber-like material behaviour.     

A Petrov–Galerkin finite element scheme for the regularized long wave equation

A Petrov-Galerkin finite element method (FEM) for the regularized long wave (RLW) equation is proposed. Finite elements are used in both the space and the time domains. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind terms in the weight functions. An implicit, conditionally stable, one-step predictor–corrector time integration scheme results. The accuracy and stability are investigated by means of local expansion by Taylor series and the resulting equivalent differential equation. An analysis based on a linear Fourier series solution and the Von Neumanns stability criterion is also performed. Based on the order of the analytical approximations and of the domain discretization it is concluded that the scheme is of third order in the nonlinear version and of fourth order in the linear version. Three numerical experiments of wave propagation are presented and their results compared with similar ones in the literature: solitary wave propagation, undular bore propagation, and cnoidal wave propagation. It is concluded that the present scheme possesses superior conservation and accuracy properties.This work has been partially supported by the Fundação para a Ciência e Tecnologia, under project POCTI/ECM/41800/2001.     

Optimization of 2D boundary element models using β-splines and genetic algorithms

The optimization of bidimensional shapes is one of the most commonly addressed problems in engineering. This work is concerned with the use of Genetic Algorithms (GAs) and β-spline-curves modeling for the optimization of Boundary Element Models (BEM). The paper briefly summarizes the basis of the GAs formulation and describes how to use refined genetic operators. The model boundary is discretized by using the BEM, and selected parts of the boundary are modeled by using β-spline curves, in order to allow easy remeshing and adaptation of the boundary to the external actions. Two numerical examples are presented and discussed in detail, showing that the proposed combined technique is able to optimize the shape of the domains with minimum computational effort. The reduction in the model area is significant, without violating the restrictions imposed to the model.     

Development of a scalable finite element solution to the Navier–Stokes equations

A scalable numerical model to solve the unsteady incompressible Navier–Stokes equations is developed using the Galerkin finite element method. The coupled equations are decoupled by the fractional-step method and the systems of equations are inverted by the Krylov subspace iterations. The data structure makes use of a domain decomposition of which each processor stores the parameters in its subdomain, while the linear equations solvers and matrices constructions are parallelized by a data parallel approach. The accuracy of the model is tested by modeling laminar flow inside a two-dimensional square lid-driven cavity for Reynolds numbers at 1,000 as well as three-dimensional turbulent plane and wavy Couette flow and heat transfer at high Reynolds numbers. The parallel performance of the code is assessed by measuring the CPU time taken on an IBM SP2 supercomputer. The speed up factor and parallel efficiency show a satisfactory computational performance.The authors wish to acknowledge Mr. W. K. Kwan of The University of Hong Kong for his help in using the IBM SP2 supercomputer.     

Simulations of nonlinear fracture process zone in cementitious material—a boundary element approach

A boundary element method based numerical model is presented to simulate the nonlinear fracture process zone in cementitious materials. A cohesive type stress-separation constitutive relationship (-w curve) is incorporated in the model to represent the process zone. Numerical algorithms for both force-controlled (prescribed loading history) and crack-tip-control-led (prescribed crack tip position) are implemented to allow whole range simulations, including strain-softening and snap-back behavior. The numerical program includes special features to permit re-adjustment of nodal points such that accurate determination of the crack-tip position is achieved. A series of numerical simulations on both 3-point beams and double cantilever beams (DCB) are conducted to investigate the development of the inelastic process zone with respect to load level, loading configuration, specimen size, and the stress-separation relationship in the process zone. Size effect on fracture resistance is clearly demonstrated. Conclusions are drawn regarding the importance of determining the details of -w curve (i.e., the values of f _t and w _c )and the need for re-evaluating the R-curves approach in cementitious materials.