Consistent mesh tying methods for topologically distinct discretized surfaces in non‐linear solid mechanics

It is known that problems arise over existence and uniqueness of solution in quasi–static contact problems involving large coefficients of Coulomb friction. This paper considers the behaviour of an elastically supported mass with three translational degrees of freedom that can make contact with a rigid Coulomb friction support. A critical coefficient of friction is identified, above which the quasi–static solution can be non–unique. A numerical solution of the corresponding dynamic problem shows that the state realized then depends on the initial conditions. Even below the critical coefficient of friction, the dynamic solution can deviate from the quasi–static even for arbitrarily small loading rates. Typical dynamic responses include limit–cycle oscillations in velocity, oscillations involving a brief period of zero velocity (stick) and ‘stick–slip’ motion in which the mass spends significant periods in a state of stick, interspersed with short rapid–slip periods. All these non–steady motions involve non–rectilinear motion, even in cases where the time derivative of the applied load is constant in direction. A perturbation analysis is performed on the quasi–static frictional slip solution and predicts instability for certain slip directions, depending on functions of the off–diagonal stiffness components and the coefficient of friction. These results are presented in dimensionless form in stability diagrams. It is also shown that quasi–static slip that is stable when there is no damping can be destabilized if the damping matrix has sufficiently large off–diagonal components.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

"Zienkiewicz, Prof. Olgierd Cecil, (18 May 1921–2 Jan. 2009), Professor and Head of Civil Engineering Department, 1961–88, and Director, Institute for Numerical Methods in Engineering, 1976–88, University of Wales at Swansea, then Professor Emeritus; UNESCO Professor of Numerical Methods in Engineering, Universidad Politécnica de Cataluña, Barcelona; J. Walter Professor of Engineering, University of Texas, Austin, 1988–98" published on by Oxford University Press.

Two higher-order compact finite difference approaches, the Hermitian and the Lax-Wendroff are examined by applying them to the viscous Burgers' equation. The difference equations obtained by the two methods were integrated in time through the third-order Strong-Stability-Preserving Runge-Kutta scheme. Absolute errors are computed by using an exact solution. The results are also compared with a second order central difference scheme. The Hermitian approach is far easier to implement. On uniform grids the Lax-Wendroff scheme produces smaller errors during the initial stages, but both methods are equally good for larger durations of integration. The convergence rate of the Hermitian scheme is slightly higher than the Lax-Wendroff scheme although both are of fourth order. Both schemes are unstable beyond a certain step size in time and space. When numerical boundary conditions are imposed, second-order conditions produce the best results whereas linear extrapolation proved to be the worst. It was also observed that large domains were required to implement the numerical boundary conditions properly. There was no detrimental effect on the accuracy of the results obtained through either of the two schemes when the size of the domain was greatly increased. Both schemes showed remarkable improvement in accuracy when clustered grids were employed. However much smaller time steps are required for stable solutions.

We present a new approach to simulating unsteady fluid flows, with only very few degrees of freedom, by employing directly eigenmodes extracted from digital particle image velocimetry experimental data. In particular, we formulate standard Galerkin and nonlinear Galerkin approximations of the incompressible Navier–Stokes equations using hierarchical empirical eigenfunctions extracted from an ensemble of flow snapshots. We demonstrate that standard Galerkin approaches produce simulations capable of capturing the short–term dynamics of the flow, but nonlinear Galerkin projections are more effective in capturing both the short– and long–term dynamics, leading to bounded solutions. These findings are documented by applying these approaches to flow past a stationary circular cylinder at Reynolds number 610.

The International Journal for Numerical Methods in Engineering keeps its first place in impact factor ranking for 2008 with the historic high impact factor of 2.229, as announced in the 2008 Journal Citation Reports® (Thomson Reuters, 2009). Copyright © 2009 John Wiley & Sons, Ltd.

Free Access Other titles from ISTE in Numerical Methods in Engineering Alain Dervieux, Alain DervieuxSearch for more papers by this authorFrédéric Alauzet, Frédéric AlauzetSearch for more papers by this authorAdrien Loseille, Adrien LoseilleSearch for more papers by this authorBruno Koobus, Bruno KoobusSearch for more papers by this author Book Author(s):Alain Dervieux, Alain DervieuxSearch for more papers by this authorFrédéric Alauzet, Frédéric AlauzetSearch for more papers by this authorAdrien Loseille, Adrien LoseilleSearch for more papers by this authorBruno Koobus, Bruno KoobusSearch for more papers by this author First published: 15 August 2022 https://doi.org/10.1002/9781394164028.scard AboutPDFPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShareShare a linkShare onFacebookTwitterLinked InRedditWechat Mesh Adaptation for Computational Fluid Dynamics 2: Unsteady and Goal-oriented Adaptation RelatedInformation

Numerical Methods in Engineering with Python, 2nd Edition is a text for engineering students and a reference for practicing engineers, especially those who wish to explore Python. This new edition features 18 additional exercises and the addition of rational function interpolation. Brent's method of root finding was replaced by Ridder's method, and the Fletcher-Reeves method of optimization was dropped in favor of the downhill simplex method. Each numerical method is explained in detail, and its shortcomings are pointed out. The examples that follow individual topics fall into two categories: hand computations that illustrate the inner workings of the method and small programs that show how the computer code is utilized in solving a problem. This second edition also includes more robust computer code with each method, which is available on the book Web site. This code is made simple and easy to understand by avoiding complex bookkeeping schemes, while maintaining the essential features of the method.

Communications in Numerical Methods in Engineering, founded by Roland W. Lewis in 1985, will change its title to the ‘International Journal for Numerical Methods in Biomedical Engineering’ and have a revised Aims and Scope. Copyright © 2009 John Wiley & Sons, Ltd.

On the use of a reduced enhanced solid-shell (RESS) element for sheet forming simulations

This work presents a methodology for free surface non-Newtonian incompressible flows using a particle position-based formulation of the Particle Finite Element Method (PFEM) [1], motivated by the simulation of fresh concrete flow. This method is based on the Lagrangian description of the flow and represents the fluid domain by cloud of particles, with a finite element mesh being built, taking the particles as nodes, to solve the motion equations and update the particle positions. To deal with large distortions of fluid flows, such mesh is constantly rebuilt, leading to a method very robust to deal with topological changes within the fluid domain. This approach requires updating the reference configuration, enabling the use of partially or fully updated Lagrangian descriptions. The weak solution is based on the stationary energy principle considering current nodal positions and pressures as variational parameters, deviating from the common practice of employing velocity and pressure as unknowns. Furthermore, smoothed versions of the Bingham and Herschel-Bulkley viscoplastic models are chosen to represent the fresh concrete behavior, keeping stresses independent of strain history and making updating the reference for the fluid simple, without the need for considering the stress distribution in the past reference. The implicit α-generalized strategy is chosen for time integration, enabling second order convergence and ensuring good stability due to the control over numerical dissipation at high frequencies. Finally, selected 2D and 3D examples are simulated to test and verify the proposed methodology.[1] IDELSOHN, S. R.; ONATE, E.; Del Pin, F. The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. International Journal for Numerical Methods in Engineering, John Wiley and Sons, Ltd, v. 61, n. 7, p. 964–989, oct 2004. ISSN 1097-0207.

Analysis and computation of a least-squares method for consistent mesh tying

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.

An interface between an hp-adaptive finite element package and the pre- and post-processor GiD

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