Numerical modelling of impulse wave generated by fast landslides

The damage caused by impulse waves generated in water bodies by fast landslides can be very high in terms of human lives and economic losses. The complex phenomena taking place in this highly unsteady process are difficult to model because three interacting phases: air, water and soil are involved. Solutions currently available are based on either closed form equations supported experimentally or the depth integrated Navier–Stokes equations. The latter, although of more general applicability, requires knowledge of the evolution of the bathimetry and slide drag forces and their applicability may be restricted by the steep slopes existing in most real cases. To avoid these limitations, the authors propose the solution of the full Navier–Stokes equations, using indicator functions to assign the material properties to each spatial point in the domain. The method performance is illustrated by comparison against the experimental results obtained in a physical model of an actual case. Copyright © 2004 John Wiley & Sons, Ltd.     

The meshless finite element method

A meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n, and the condition that the shape functions depend only on the node positions. Furthermore, the method proposed also shares several of the advantages of the finite element method such as: (a) the simplicity of the shape functions in a large part of the domain; (b) C⁰ continuity between elements, which allows the treatment of material discontinuities, and (c) ease of introduction of the boundary conditions. Copyright © 2003 John Wiley & Sons, Ltd.     

Analysis of time varying errors in quadratic finite element approximation of hyperbolic problems

A methodology for analysing the numerical errors generated by schemes using high-order approximation is presented. Based on Fourier analysis, this methodology is illustrated through the study of the θ-weighting Taylor–Galerkin finite element model applied to an unsteady one-dimension advection problem with quadratic elements. Results show that the dissipation and dispersion errors may be computed by considering simultaneously the so-called physical and computational modes and then, contrarily to what is shown when linear approximation is considered, these errors present a transient behaviour. Moreover, it appears that the errors computed at the end node and at the middle node present in general a different behaviour which in some cases may be opposed to one another. Numerical tests are presented to support the validity of the proposed strategy. We recommend strongly the use of this method for studying the behaviour of numerical schemes based on high-order approximations.     

Numerical analysis of a new Eulerian–Lagrangian finite element method applied to steady‐state hot rolling processes

A finite element code for steady‐state hot rolling processes of rigid–visco‐plastic materials under plane–strain conditions was developed in a mixed Eulerian–Lagrangian framework. This special set up allows for a direct calculation of the local deformations occurring at the free surfaces outside the contact region between the strip and the work roll. It further simplifies the implementation of displacement boundary conditions, such as the impenetrability condition. When applied to different practical hot rolling situations, ranging from thick slab to ultra‐thin strip rolling, the velocity–displacement based model (briefly denoted as vu‐model) in this mixed Eulerian–Lagrangian reference system proves to be a robust and efficient method. The vu‐model is validated against a solely velocity‐based model (vv‐model) and against elementary methods based on the Kármán–Siebel and Orowan differential equations. The latter methods, when calibrated, are known to be in line with experimental results for homogeneous deformation cases. For a massive deformation it is further validated against the commercial finite‐element software package Abaqus/Explicit. It is shown that the results obtained with the vu‐model are in excellent agreement with the predictions of the vv‐model and that the vu‐model is even more robust than its vv‐counterpart. Throughout the study we assumed a rigid cylindrical work roll; only for the homogeneous test case, we also investigated the effect of an elastically deformable work roll within the frame of the Jortner Green's function method. The new modelling approach combines the advantages of conventional Eulerian and Lagrangian modelling concepts and can be extended to three dimensions in a straightforward manner. Copyright © 2004 John Wiley & Sons, Ltd.     

A new stabilized finite element method for reaction–diffusion problems: The source‐stabilized Petrov–Galerkin method

This paper proposes a new stabilized finite element method to solve singular diffusion problems described by the modified Helmholtz operator. The Galerkin method is known to produce spurious oscillations for low diffusion and various alternatives were proposed to improve the accuracy of the solution. The mostly used methods are the well‐known Galerkin least squares and Galerkin gradient least squares (GGLS). The GGLS method yields the exact nodal solution in the one‐dimensional case and for a uniform mesh. However, the behavior of the method deteriorates slightly in the multi‐dimensional case and for non‐uniform meshes. In this work we propose a new stabilized finite element method that leads to improved accuracy for multi‐dimensional problems. For the one‐dimensional case, the new method leads to the same results as the GGLS method and hence provides exact nodal solutions to the problem on uniform meshes. The proposed method is a Galerkin discretization used to solve a modified equation that includes a term depending on the gradient of the original partial differential equation. Copyright © 2008 John Wiley & Sons, Ltd.     

Fully automatic modelling of mixed-mode crack propagation using scaled boundary finite element method

The newly-developed scaled boundary finite element method (SBFEM) is able to calculate stress intensity factors directly because the singularity in stress solutions at crack tips is analytically represented. By taking this advantage, a mixed-mode crack propagation model based on linear elastic fracture mechanics (LEFM) was developed in this study. A domain is first divided into a few subdomains. Because the dimensions and shapes of subdomains can be flexibly varied and only the domain boundaries or common edges between subdomains are discretised in the SBFEM, a remeshing procedure as simple as in boundary element methods was developed with minimum mesh changes whereas the generality and flexibility of the FEM is well maintained. Fully-automatic modelling of mixed-mode crack propagation is then achieved by combining the remeshing procedure with a propagation criterion. Three mixed-mode examples were modelled. Comparisons of the numerical results with those from available publications show that the developed model is capable of predicting crack trajectories and load-displacement relations accurately and efficiently.     

A finite element method for light activated shape‐memory polymers

Shape‐memory polymers (SMPs) belong to a class of smart materials that have shown promise for a wide range of applications. They are characterized by their ability to maintain a temporary deformed shape and return to an original parent permanent shape. In this paper, we consider the coupled photomechanical behavior of light activated shape‐memory polymers (LASMPs), focusing on the numerical aspects for finite element simulations at the engineering scale. The photomechanical continuum framework is summarized, and some specific constitutive equations for LASMPs are described. Numerical implementation of the multiphysics governing partial differential equations takes the form of a user defined element subroutine within the commercial software package ABAQUS . We verify our two‐dimensional and three‐dimensional finite element procedure for multiple analytically tractable cases. To show the robustness of the numerical implementation, simulations are performed under various geometries and complex photomechanical loading. Copyright © 2016 John Wiley & Sons, Ltd.     

A Fourier analysis and dynamic optimization of the Petrov–Galerkin finite element method

A Fourier analysis of the linear and quadratic N + 1 and N + 2 Petrov–Galerkin finite element methods applied to the one-dimensional transient convective-diffusion equation is performed. The results show that a priori optimization of the N + 1 method is not possible because dissipative errors are introduced as dispersive errors are reduced (any optimization is subjective). However, a priori optimization of the N + 2 Petrov–Galerkin method is possible because the reduction of dispersion errors can be accomplished without the addition of artificial dissipation. The Spectrally Weighted Average Phase Error Method (SWAPEM) for the optimization of the N + 2 Petrov–Galerkin method is introduced, in which the N + 2 weighting parameter is chosen at each time step to minimize the integral over wave number of the phase error of Fourier modes, weighted by the frequency content of the global solution at the previous time step (obtained via FFT). The method is dynamic, and general in that the dependence of the weighting parameter on the solution waveform is accounted for. Optimal values predicted by the method are in excellent agreement with those suggested by the numerical experimentation of others. Simulations of the pure convective transport of a Gaussian plume and a triangle wave are discussed to illustrate the effectiveness of the method.     

A mixed finite element method for non‐linear and nearly incompressible elasticity based on biorthogonal systems

We present a finite element method for non‐linear and nearly incompressible elasticity. The formulation is based on Petrov–Galerkin discretization for the pressure and is closely related to the average nodal pressure formulation presented earlier in the context of incompressible and nearly incompressible dynamic explicit applications (Commun. Numer. Meth. Engng 1998; 14 :437–449). Some numerical examples are presented to show the efficiency of the approach. Copyright © 2009 John Wiley & Sons, Ltd.     

Delamination modelling of GLARE using the extended finite element method

In this paper, an application of the Extended Finite Element Method (XFEM) for simulation of delamination in fibre metal laminates is presented. The study consider a double cantilever beam made of fibre metal laminate in which crack opening in mode I and crack propagation were studied. Comparison with the solution by standard Finite Element Method (FEM) as well as with experimental tests is provided. To the authors’ knowledge, this is the first time that XFEM is used in the fracture analysis of fibre metal laminates such as GLARE. The results indicated that XFEM could be a promising technique for the failure analysis of composite structures.     

Imposing Dirichlet boundary conditions in the extended finite element method

This paper is devoted to the imposition of Dirichlet‐type conditions within the extended finite element method (X‐FEM). This method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces. Imposing Neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh (observed numerically in earlier papers). On the contrary, much less work has been devoted to Dirichlet boundary conditions for the X‐FEM (or the limiting case of stiff boundary conditions). In this paper, we introduce a strategy to impose Dirichlet boundary conditions while preserving the optimal rate of convergence. The key aspect is the construction of the correct Lagrange multiplier space on the boundary. As an application, we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding (RTM) process while avoiding remeshing. The case of inner conditions is also discussed as well as two important practical cases: material interfaces and phase‐transformation front capturing. Copyright © 2006 John Wiley & Sons, Ltd.     

The improvement of crack propagation modelling in triangular 2D structures using the extended finite element method

In this paper, a novel geometric method combined with the piecewise linear function method is introduced into the extended finite element method (XFEM) to determine the crack tip element and crack surface element. Then, by combining with the advanced mesh technique, a novel method is proposed to improve the modelling of crack propagation in triangular 2D structure with the XFEM. The numerical tests show that the accuracy, the convergence, and the stability of the XFEM can be improved using the proposed method. Moreover, the applicability of the conventional multiple enrichment scheme is discussed. Compared with the proposed method, the conventional multiple enrichment scheme has deficiency in mixed mode I and II crack. Finally, a comparative study shows that the performance of the XFEM by using the proposed method to model the crack propagation can be greatly improved.     

Application of the level set method to the finite element solution of two‐phase flows

This paper presents a method to solve two‐phase flows using the finite element method. On one hand, the algorithm used to solve the Navier–Stokes equations provides the neccessary stabilization for using the efficient and accurate three‐node triangles for both the velocity and pressure fields. On the other hand, the interface position is described by the zero‐level set of an indicator function. To maintain accuracy, even for large‐density ratios, the pseudoconcentration function is corrected at the end of each time step using an algorithm successfully used in the finite difference context. Coupling of both problems is solved in a staggered way. As demonstrated by the solution of a number of numerical tests, the procedure allows dealing with problems involving two interacting fluids with a large‐density ratio. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical modeling of corrosion pit propagation using the combined extended finite element and level set method

A sharp-interface Eulerian formulation for modeling the propagation of localized pitting corrosion is presented. This formulation allows for an accurate representation of the corrosion front independent of the underlying finite element mesh and handles complex morphological transitions such as pit merging without requiring remeshing or mesh-moving procedures. First, the governing equations of the moving interface problem associated with pit growth are derived for the two-phase (metal-solution) system. Next, the implementation of the combined extended finite element and level set method and the procedure to enforce interface conditions are discussed. Finally, the method is validated by conducting several benchmark numerical studies, and comparing the results with published experimental data and existing numerical studies. Simulation studies indicate that: during diffusion-controlled corrosion an isolated pit grows as a semi-circular shape, whereas closely spaced pits merge and grow into an elongated elliptical shape; and only during activation-controlled corrosion the initial pit morphology is preserved.     

Nonlocal damage modelling using the element-free Galerkin method in the frame of finite strains

Prediction of size effects has been a challenging problem since some experiments found the size effect in material damage. Both material model and numerical algorithm have to be improved to consider the complex damage process. In the present paper we implement element-free Galerkin (EFG) method for a strain-gradient based nonlocal damage model and use it to analyze ductile material damage process. The EFG algorithm overcomes some drawbacks of the FEM in convergence of numerical iteration due to large deformations as well as evaluation of the higher-order gradients of the plastic strain. The numerical benchmarks show that the EFG method for the nonlocal damage model provides more stable numerical results. The size effect in notched specimens can be predicted in the computations. Both ductile fracture in tensile specimens as well as their size effects are investigated and the computational results agree very well with experiments.     

Coupling of a non‐overlapping domain decomposition method for a nodal finite element method with a boundary element method

Non‐overlapping domain decomposition techniques are used to solve the finite element equations and to couple them with a boundary element method. A suitable approach dealing with finite element nodes common to more than two subdomains, the so‐called cross‐points, endows the method with the following advantages. It yields a robust and efficient procedure to solve the equations resulting from the discretization process. Only small size finite element linear systems and a dense linear system related to a simple boundary integral equation are solved at each iteration and each of them can be solved in a stable way. We also show how to choose the parameter defining the augmented local matrices in order to improve the convergence. Several numerical simulations in 2D and 3D validating the treatment of the cross‐points and illustrating the strategy to accelerate the iterative procedure are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

The partition of unity finite element method for elastic wave propagation in Reissner–Mindlin plates

This paper reports a numerical method for modelling the elastic wave propagation in plates. The method is based on the partition of unity approach, in which the approximate spectral properties of the infinite dimensional system are embedded within the space of a conventional finite element method through a consistent technique of waveform enrichment. The technique is general, such that it can be applied to the Lagrangian family of finite elements with specific waveform enrichment schemes, depending on the dominant modes of wave propagation in the physical system. A four‐noded element for the Reissner–Mindlin plate is derived in this paper, which is free of shear locking. Such a locking‐free property is achieved by removing the transverse displacement degrees of freedom from the element nodal variables and by recovering the same through a line integral and a weak constraint in the frequency domain. As a result, the frequency‐dependent stiffness matrix and the mass matrix are obtained, which capture the higher frequency response with even coarse meshes, accurately. The steps involved in the numerical implementation of such element are discussed in details. Numerical studies on the performance of the proposed element are reported by considering a number of cases, which show very good accuracy and low computational cost. Copyright © 2006 John Wiley & Sons, Ltd.     

Analysis of three‐dimensional crack initiation and propagation using the extended finite element method

An Erratum has been published for this article in International Journal for Numerical Methods in Engineering 2005, 63(8): 1228. We present a new formulation and a numerical procedure for the quasi‐static analysis of three‐dimensional crack propagation in brittle and quasi‐brittle solids. The extended finite element method (XFEM) is combined with linear tetrahedral elements. A viscosity‐regularized continuum damage constitutive model is used and coupled with the XFEM formulation resulting in a regularized ‘crack‐band’ version of XFEM. The evolving discontinuity surface is discretized through a C⁰ surface formed by the union of the triangles and quadrilaterals that separate each cracked element in two. The element's properties allow a closed form integration and a particularly efficient implementation allowing large‐scale 3D problems to be studied. Several examples of crack propagation are shown, illustrating the good results that can be achieved. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite element method for poro mechanical modelling of geotechnical problems using local second gradient models

In this paper, a new finite element method is described and applied. It is based on a theory developed to model poromechanical problems where the mechanical part is obeying a second gradient theory. The aim of such a work is to properly model the post localized behaviour of soils and rocks saturated with a pore fluid. Beside the development of this new coupled theory, a corresponding finite element method has been developed. The elements used are based on a weak form of the relation between the deformation gradient and the second gradient, using a field of Lagrange multipliers. The global problem is solved by a system of equations where the kinematic variables are fully coupled with the pore pressure. Some numerical experiments showing the effectiveness of the method ends the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

A high‐order approach for modelling transient wave propagation problems using the scaled boundary finite element method

A high‐order time‐domain approach for wave propagation in bounded and unbounded domains is proposed. It is based on the scaled boundary FEM, which excels in modelling unbounded domains and singularities. The dynamic stiffness matrices of bounded and unbounded domains are expressed as continued‐fraction expansions, which leads to accurate results with only about three terms per wavelength. An improved continued‐fraction approach for bounded domains is proposed, which yields numerically more robust time‐domain formulations. The coefficient matrices of the corresponding continued‐fraction expansion are determined recursively. The resulting solution is suitable for systems with many DOFs as it converges over the whole frequency range, even for high orders of expansion. A scheme for coupling the proposed improved high‐order time‐domain formulation for bounded domains with a high‐order transmitting boundary suggested previously is also proposed. In the time‐domain, the coupled model corresponds to equations of motion with symmetric, banded and frequency‐independent coefficient matrices, which can be solved efficiently using standard time‐integration schemes. Numerical examples for modal and time‐domain analysis are presented to demonstrate the increased robustness, efficiency and accuracy of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.