Convergence analysis of a finite volume method via a new nonconforming finite element method

The article is devoted to the study of convergence properties of a Finite Volume Method (FVM) using Voronoi boxes for discretization. The approach is based on the construction of a new nonconforming Finite Element Method (FEM), such that the system of linear equations coincides completely with that for the FVM. Thus, by proving convergence properties of the FEM, we obtain similar ones of the FVM. In this article, the investigations are restricted to the Poisson equation. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:213–231, 1998     

Study of Mode Conversion at Defects in Rope Structures using Ultrasonic Waves

Ultrasonic waves travel in rope structures over long distances as guided waves, allowing for effective health monitoring. In order to localize and characterize defects, an exact knowledge of the propagation, reflection, and transmission properties of the ultrasonic waves is required. These properties can be obtained using the Finite Element Method by modeling a segment of the periodic waveguide with a periodicity condition. The solution of the corresponding eigenvalue problem leads to all propagating modes of the waveguide as well as locally generated evanescent modes. The Boundary Element Method (BEM) is used in combination with the Finite Element Method for characterizing the wave propagation. The mode conversion at discontinuities, such as cracks or notches, can be subsequently described by reflection and transmission coefficients. The simulation results are the corresponding coefficients as a function of frequency and enable the selection of adequate modes for an effective defect detection. Additionally, it is demonstrated that along with the localization of cracks, conclusions about the crack geometry can be made with the help of reflection and transmission coefficients. The reliability and numerical accuracy of the simulation results are verfied by comparison with experimental findings. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lamb wave propagation using Wave Finite Element Method

Some of the available techniques for Lamb wave propagation simulation are the Finite Element Method (FEM), the Boundary Element Method and the Finite Difference Method. The FEM is the best method when complex damage, geometry or boundary is involved. However, high Lamb wave frequency requires very small element size thus high computational cost in FEM analysis. By using the existence of periodicity in plates, an attempt to reduce this computational cost is done using Wave FEM. The applicability of this method to model Lamb wave propagation in plate is first assessed in this paper for the 1-D wave propagation and compared with FEM explicit method. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fuzzy logic control of electrodynamic levitation devices coupled to dynamic finite volume method analysis

Electrodynamic levitation devices, which utilizing eddy currents induced in the levitated item to produce the repulsive force, are being involved in many engineering applications due to its fast response. This kind of repulsion is particularly used in electromagnetic launcher, electromagnetic brake and other applications. To analyze and improve the dynamic behavior and performances of such devices, the conventional way is using the finite element method (FEM), due to its ability of using adaptive mesh to handle complex geometries. Nevertheless, it has a serious limitation in efficiency for large number of variables which is reflected by the high cost in terms of computational properties. During the past few years, the finite volume method FVM formulations have gained attention inside the electromagnetic community. The method has been proved its effectiveness in the solution of different kinds of problems, such as in magnetostatic field computation and eddy current nondestructive testing. The FVM method is particularly attractive thanks to its small required storage memory and reduced CPU time. In this paper an FVM model is developed to analyze the dynamic characteristic of the motion of the electrodynamic levitation device TEAM Workshop Problem 28. The dynamic characteristic of the motion is obtained by solving the electromagnetic equation coupled to the mechanical one. The repulsive force applied to the levitated plate of TEAM Workshop Problem 28, is computed by the interaction between eddy current induced in the plate and the magnetic flux density. A comparison between experimental and numerical results is carried out to show the efficiency of the developed model. What’s more, based on the developed FVM model, a fuzzy logic controller FLC is designed and implemented to control the position of the levitated item.     

Konvergenzanalysis einer FVM für Konvektions‐Diffusions‐Probleme

The paper is devoted to the convergence analysis of a well‐known cell‐centered Finite Volume Method (FVM) for a convection‐diffusion‐problem in R².     

High resolution multi-moment finite volume method for supersonic combustion on unstructured grids

In this study, we present a novel numerical model for simulating detonation waves on unstructured grids. In contrast to the conventional finite volume method (FVM), two types of moment comprising the volume-integrated average (VIA) and the point value (PV) at the cell vertex are treated as the evolution variables for the reacting Euler equations. The VIA is computed based on a finite volume formulation of the flux form where the conventional Riemann problem is solved by the HLLC Riemann solver. The PV is updated in a point-wise manner by using the differential formulation where the Roe solver is used to compute the differential Riemann problems. In order to increase the accuracy around discontinuities, numerical oscillations and dissipations are reduced using the boundary variation diminishing algorithm. Convergence tests demonstrated that the proposed model could achieve third-order accuracy with unstructured grids for reacting Euler equations. The high resolution property of the proposed method was verified based on simulations of several detonation wave propagation problems in two and three dimensions. In particular, the current model could resolve the cellular structures with fewer degrees of freedom for the unstable oblique detonation wave problem. These fine structures may be smoothed out by the conventional FVM due to the excessive amount of numerical dissipation errors. Importantly, a simulation of stiff detonation waves showed that the proposed method could capture the correct position of the reaction front whereas the conventional FVMs produced spurious phenomena. Thus, the proposed model can obtain highly accurate solutions for detonation problems on unstructured grids, which is highly advantageous for real applications involving complex geometrical configurations.     

Computation of crack propagation in 3-dimensional anisotropic solids

This contribution presents ideas, how crack propagation in three-dimensional solids composed of anisotropic materials can be predicted using the Griffith energy principle. Since the work of Irwin the change of potential energy caused by a straight elongation of a crack in an isotropic two-dimensional homogeneous structure can be expressed in quadratic terms of the stress intensities at the crack tip. This result was generalized in the last decades using methods of asymptotic analysis by many authors [1] to more complicated geometries, to anisotropic and inhomogeneous materials. With the energy release rate at hand, quasi-static scenarios of crack propagation can be simulated for plane problems [2], but this is still a complicated task for three-dimensional problems [3]. We show an idea how the change of energy caused by propagation of a crack surface in a fully three-dimensional solid of nearly arbitrary shape can be computed in anisotropic materials. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal balance between mass and smoothed stiffness in simulation of acoustic problems

The Finite Element Method (FEM) is known to behave overly-stiff, which leads to an imbalance between the mass and stiffness matrices within discretized systems. In this work, for the first time, a model is developed that provides optimal balance between discretized mass and smoothed stiffness—the mass-redistributed alpha finite element method (MR-αFEM). This new method improves on the computational efficiency of the FEM and Smoothed Finite Element Methods (S-FEM). The rigorous research conducted ensures that stiffness with the parameter, α, optimally matches the mass with a flexible integration point, q. The optimal balance system significantly reduces the dispersion error of acoustic problems, including those of single and multi-fluids in both time and frequency domains. The excellent properties of the proposed MR-αFEM are validated using theoretical analyses and numerical examples.     

Out-of-plane displacement measurement near the crack tip during a crack propagation: Validation of a 3D formulation for a specimen in PMMA loaded in mode I.

The fundamental aim of this study is the determination zone of the 3D effects and the transient one at the vicinity of the crack tip during a crack propagation in brittle materials ( PMMA ) using an optical method (Michelson interferometer). With the obtained interferograms, we can extract the phase (thus the relief) by using a new numerical approach based on the principle of images correlation between real fringes and virtual fringes. Different dynamic tests are realized by a plate loaded in mode I under a constant loading. We compare the obtained data with the two-dimensional theory of Westergaard (plane stress hypothesis) [1]. With the divergence is established, we propose a new 3D formulation, based on a formulation employed for static crack, which takes into account 3D and transient effects. For the static cracks, the 3D effects relate to a presence of the state of three-dimensional stresses. However in dynamics, the transient effects appear and are related to the crack propagation velocity. The 3D effects and transient effects lead to results equivalent to experimental ones in terms of displacement but are completely different to results given by the two-dimensional theory near the crack tip. It is possible to quantify the zone when the plane stress hypothesis is not valid according to the crack propagation speed V. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hypersingularities in three-dimensional crack configurations in composite laminates

Three-dimensional crack configurations in composite laminates are studied by means of the Scaled Boundary Finite Element Method (SBFEM) particularly regarding stress singularities and their associated deformation modes. The SBFEM is an efficient semi-analytical method that permits solving linear elastic mechanical problems. Only the boundary needs to be discretized while the problem is considered analytically in the direction of the dimensionless radial coordinate pointing from the scaling center to the boundary . An important advantage is that it requires no additional effort for the characterization of existing stress singularities. The situation of two meeting inter-fiber cracks is investigated in detail, considering different materials and fiber / crack orientations. It is shown that in three-dimensional crack configurations in composite laminates so-called hypersingularities can occur, i.e. stress singularities which are even stronger than the classical crack singularity. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Study on the dynamics of the double cantilever-beam specimen of decagonal Al-Ni-Co quasicrystals

This paper investigated dynamic initiation of crack growth and crack fast propagation for the double cantilever-beam specimen (DCB) of two-dimensional decagonal Al-Ni-Co quasicrystals. The elasto-/hydro-dynamic model for wave propagation and diffusion together with their interaction is adopted. Numerical results of stresses, displacements and dynamic stress intensity factor are obtained by the finite difference method. Dynamic initiation of crack growth and crack fast propagation are discussed in detail in which the latter is a nonlinear problem arising from moving boundary effect, which is in particular explored.     

Dynamic Wave Propagation in Porous Media Semi-Infinite Domains

The problem of dynamic wave propagation in semi-infinite domains is of great importance, especially, in subjects of applied mechanics and geomechanics, such as the issues of earthquake wave propagation in an infinite half-space and soil-structure interaction under seismic loading. In such problems, the elastic waves are supposed to propagate to infinity, which requires a special treatment of the boundaries in initial boundary-value problems (IBVP). Saturated porous materials, e. g. soil, basically represent volumetrically coupled solid-fluid aggregates. Based on the continuum-mechanical principles and the established macroscopic Theory of Porous Media (TPM) [1, 2], the governing balance equations yield a coupled system of partial differential equations (PDE). Restricting the discussion to the isothermal and geometrically linear case, this system comprises the solid and fluid momentum balances and the overall volume balance, and can be conveniently treated numerically following an implicit monolithic approach [3]. Therefore, the equations are firstly discretised in space using the mixed Finite Element Method (FEM) together with quasi-static Infinite Elements (IE) at the boundaries that represent the extension of the domain to infinity [4], and secondly in time using an appropriate implicit time-integration scheme. Additionally, a stable implementation of the Viscous Damping Boundary (VDB) method [5] for the simulation of transient waves at infinity is presented, which implicitly treats the damping boundary terms in a weakly imposed sense. The proposed algorithm is implemented into the FE tool PANDAS and tested on a two-dimensional IBVP. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a numerical scheme for curved crack propagation based on configurational forces and maximum dissipation

A numerical scheme is presented to predict crack trajectories in two dimensional components. First a relation between the curvature in mixed–mode crack propagation and the corresponding configurational forces is derived, based on the principle of maximum dissipation. With the help of this, a numerical scheme is presented which is based on a predictor–corrector method using the configurational forces acting on the crack together with their derivatives along real and test paths. With the help of this scheme it is possible to take bigger than usual propagation steps, represented by splines. Essential for this approach is the correct numerical determination of the configurational forces acting on the crack tip. The methods used by other authors are shortly reviewed and an approach valid for arbitrary non–homogenous and non–linear materials with mixed–mode cracks is presented. Numerical examples show, that the method is a able to predict the crack paths in components with holes, stiffeners etc. with good accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient Simulation of Wave Propagation with Implicit Finite Difference Schemes

Finite difference method is an important methodology in the approximation of waves. In this paper, we will study two implicit finite difference schemes for the simulation of waves. They are the weighted alternating direction implicit (ADI) scheme and the locally one-dimensional (LOD) scheme. The approximation errors, stability conditions, and dispersion relations for both schemes are investigated. Our analysis shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme. Moreover, the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time. In order to improve computational efficiency, numerical algorithms based on message passing interface (MPI) are implemented. Numerical examples of wave propagation in a three-layer model and a standard complex model are presented. Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.     

Augmented high order finite volume element method for elliptic PDEs in non-smooth domains: Convergence study

The accuracy of a finite element numerical approximation of the solution of a partial differential equation can be spoiled significantly by singularities. This phenomenon is especially critical for high order methods. In this paper, we show that, if the PDE is linear and the singular basis functions are homogeneous solutions of the PDE, the augmentation of the trial function space for the Finite Volume Element Method (FVEM) can be done significantly simpler than for the Finite Element Method. When the trial function space is augmented for the FVEM, all the entries in the matrix originating from the singular basis functions in the discrete form of the PDE are zero, and the singular basis functions only appear in the boundary conditions. That is to say, there is no need to integrate the singular basis functions over the elements and the sparsity of the matrix is preserved without special care. FVEM numerical convergence studies on two-dimensional triangular grids are presented using basis functions of arbitrary high order, confirming the same order of convergence for singular solutions as for smooth solutions.     

A quasi-optimal convergence result for fracture mechanics with XFEM

The aim of this Note is to give a convergence result for a variant of the eXtended Finite Element Method (XFEM) on cracked domains using a cut-off function to localize the singular enrichment area. The difficulty is caused by the discontinuity of the displacement field across the crack, but we prove that a quasi-optimal convergence rate holds in spite of the presence of elements cut by the crack. The global linear convergence rate is obtained by using an enriched linear finite element method. To cite this article: E. Chahine et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).