Deformation Driven Homogenization of Fracturing Solids

The paper discusses numerical formulations of the homogenization for solids with discrete crack development. We focus on multi–phase microstructures of heterogeneous materials, where fracture occurs in the form of debonding mechanisms as well as matrix cracking. The definition of overall properties critically depends on the developing discontinuities. To this end, we extend continuous formulations [1] to microstructures with discontinuities [2]. The basic underlying structure is a canonical variational formulation in the fully nonlinear range based on incremental energy minimization. We develop algorithms for numerical homogenization of fracturing solids in a deformation–driven context with non–trivial formulations of boundary conditions for (i) linear deformation and (ii) uniform tractions. The overall response of composite materials with fracturing microstructures are investigated. As a key result, we show the significance of the proposed non–trivial formulation of a traction–type boundary condition in the deformation–driven context. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of the spread of a viscous fluid using a bidimensional shallow water model

In this paper we propose a numerical method to solve the Cauchy problem based on the viscous shallow water equations in an horizontally moving domain. More precisely, we are interested in a flooding and drying model, used to modelize the overflow of a river or the intrusion of a tsunami on ground. We use a nonconservative form of the two-dimensional shallow water equations, in eight velocity formulation and we build a numerical approximation, based on the Arbitrary lagrangian eulerian formulation, in order to compute the solution in the moving domain.     

Development of a finite element model of metal powder compaction process at elevated temperature

This paper presents the finite element modelling of metal powder compaction process at elevated temperature. In the modelling, the behaviour of powder is assumed to be rate independent thermo-elastoplastic material where the material constitutive laws are derived based on a continuum mechanics approach. The deformation process of metal powder has been described by a large displacement based finite element formulation. The Elliptical Cap yield model has been used to represent the deformation behaviour of the powder mass during the compaction process. This yield model was tested and found to be appropriate to represent the compaction process. The staggered-incremental-iterative solution strategy has been established to solve the non-linearity in the systems of equations. Some numerical simulation results were validated through experimentation, where a good agreement was found between the numerical simulation results and the experimental data.     

Higher order unfitted FEM for Stokes interface problems

We consider the discretization of a stationary Stokes interface problem in a velocity-pressure formulation. The interface is described implicitly as the zero level of a scalar function as it is common in level set based methods. Hence, the interface is not aligned with the mesh. An unfitted finite element discretization based on a Taylor-Hood velocity-pressure pair and an XFEM (or CutFEM) modification is used for the approximation of the solution. This allows for the accurate approximation of solutions which have strong or weak discontinuities across interfaces which are not aligned with the mesh. To arrive at a consistent, stable and accurate formulation we require several additional techniques. First, a Nitsche-type formulation is used to implement interface conditions in a weak sense. Secondly, we use the ghost penalty stabilization to obtain an inf-sup stable variational formulation. Finally, for the highly accurate approximation of the implicitly described geometry, we use a combination of a piecewise linear interface reconstruction and a parametric mapping of the underlying mesh. We introduce the method and discuss results of numerical examples. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase Field Modeling of Brittle and Ductile Fracture

The phase field modeling of brittle fracture was a topic of intense research in the last few years and is now well-established. We refer to the work [1-3], where a thermodynamically consistent framework was developed. The main advantage is that the phase-field-type diffusive crack approach is a smooth continuum formulation which avoids the modeling of discontinuities and can be implemented in a straightforward manner by multi-field finite element methods. Therefore complex crack patterns including branching can be resolved easily. In this paper, we extend the recently outlined phase field model of brittle crack propagation [1-3] towards the analysis of ductile fracture in elastic-plastic solids. In particular, we propose a formulation that is able to predict the brittle-to-ductile failure mode transition under dynamic loading that was first observed in experiments by Kalthoff and Winkler [4]. To this end, we outline a new thermodynamically consistent framework for phase field models of crack propagation in ductile elastic-plastic solids under dynamic loading, develop an incremental variational principle and consider its robust numerical implementation by a multi-field finite element method. The performance of the proposed phase field formulation of fracture is demonstrated by means of the numerical simulation of the classical Kalthoff-Winkler experiment that shows the dynamic failure mode transition. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the formulation of invariant imbedding problems

The formulation of an invariant imbedding problem from a given linear, two-point boundary-value problem is not unique. In this paper, we illustrate how the formulation of the problem by partitioning the original vectory(z) into [u(z),v(z)], can affect the numerical accuracy. In fact, the partitioning, the choice of theR, O system orS, T system of equations in Scott's method, the location and number of switch points, and the switching procedure, all influence the numerical results and the ease of obtaining solutions. A new method of switching and the appropriate formulas are described, namely, the repeated switching from theR, Q system to theR, Q system of equations or from theS, T system to theS, T system of equations.     

Modeling of large deformation frictional contact in powder compaction processes

In this paper, the computational aspects of large deformation frictional contact are presented in powder forming processes. The influence of powder–tool friction on the mechanical properties of the final product is investigated in pressing metal powders. A general formulation of continuum model is developed for frictional contact and the computational algorithm is presented for analyzing the phenomena. It is particularly concerned with the numerical modeling of frictional contact between a rigid tool and a deformable material. The finite element approach adopted is characterized by the use of penalty approach in which a plasticity theory of friction is incorporated to simulate sliding resistance at the powder–tool interface. The constitutive relations for friction are derived from a Coulomb friction law. The frictional contact formulation is performed within the framework of large FE deformation in order to predict the non-uniform relative density distribution during large deformation of powder die pressing. A double-surface cap plasticity model is employed together with the nonlinear contact friction behavior in numerical simulation of powder material. Finally, the numerical schemes are examined for efficiency and accuracy in modeling of several powder compaction processes.     

Optimizing Preventive Maintenance Models

We deal with the problem of scheduling preventive maintenance (PM) for a system so that, over its operating life, we minimize a performance function which reflects repair and replacement costs as well as the costs of the PM itself. It is assumed that a hazard rate model is known which predicts the frequency of system failure as a function of age. It is also assumed that each PM produces a step reduction in the effective age of the system. We consider some variations and extensions of a PM scheduling approach proposed by Lin et al. [6]. In particular we consider numerical algorithms which may be more appropriate for hazard rate models which are less simple than those used in [6] and we introduce some constraints into the problem in order to avoid the possibility of spurious solutions. We also discuss the use of automatic differentiation (AD) as a convenient tool for computing the gradients and Hessians that are needed by numerical optimization methods. The main contribution of the paper is a new problem formulation which allows the optimal number of occurrences of PM to be determined along with their optimal timings. This formulation involves the global minimization of a non-smooth performance function. In our numerical tests this is done via the algorithm DIRECT proposed by Jones et al. [19]. We show results for a number of examples, involving different hazard rate models, to give an indication of how PM schedules can vary in response to changes in relative costs of maintenance, repair and replacement. Part of this work was carried out while the first author was a Visiting Professor in the Department of Mechanical Engineering at the University of Alberta in December 2003.     

Barotropic-Baroclinic Formulation of the Primitive Equations of the Ocean

In this article, we present an equivalent barotropic-baroclinic formulation of the primitive equations (PEs) of the ocean given in [J.L. Lions, R. Temam and S. Wang (<citeref rid="bib5">1992</citeref>). On the equations of large-scale ocean. Nonlinearity, 5, 1007-1053.]. From the numerical point of view, the main advantage of this new formulation is that the incompressibility condition appearing in the PEs in [J.L. Lions, R. Temam and S. Wang (<citeref rid="bib5">1992</citeref>). On the equations of large-scale ocean. Nonlinearity, 5, 1007-1053.] is automatically satisfied without being explicitly imposed at any stage. Some numerical schemes for the time integration of the PEs are presented and their numerical stability is discussed. These schemes are reminiscent of other schemes that have been used for other equations in particular the Navier-Stokes equations. We end the article by presenting numerical simulations of a wind-driven ocean model using the new formulation. More extensive numerical simulations and physical aspects will be presented elsewhere.     

Finite element analysis of the vibration problem of a plate coupled with a fluid

Summary. We consider the approximation of the vibration modes of an elastic plate in contact with a compressible fluid. The plate is modelled by Reissner-Mindlin equations while the fluid is described in terms of displacement variables. This formulation leads to a symmetric eigenvalue problem. Reissner-Mindlin equations are discretized by a mixed method, the equations for the fluid with Raviart-Thomas elements and a non conforming coupling is used on the interface. In order to prove that the method is locking free we consider a family of problems, one for each thickness , and introduce appropriate scalings for the physical parameters so that these problems attain a limit when . We prove that spurious eigenvalues do not arise with this discretization and we obtain optimal order error estimates for the eigenvalues and eigenvectors valid uniformly on the thickness parameter t. Finally we present numerical results confirming the good performance of the method. Received February 4, 1998 / Revised version received May 26, 1999 / Published online June 21, 2000     

A kinetic scheme for unsteady pressurised flows in closed water pipes

The aim of this paper is to present a kinetic numerical scheme for the computations of transient pressurised flows in closed water pipes. Firstly, we detail the mathematical model written as a conservative hyperbolic partial differentiel system of equations, and then we recall how to obtain the corresponding kinetic formulation. Then we build the kinetic scheme ensuring an upwinding of the source term due to the topography performed in a close manner described by Perthame and Simeoni (2001) [1] and Botchorishvili et al. (2003) [2] using an energetic balance at microscopic level. The validation is lastly performed in the case of a water hammer in an uniform pipe: we compare the numerical results provided by an industrial code used at EDF-CIH (France), which solves the Allievi equation (the commonly used equation for pressurised flows in pipes) by the method of characteristics, with those of the kinetic scheme. It appears that they are in a very good agreement.     

Some Remarks on a Vibro-Impact Scheme

We consider a mechanical system subject to a unilateral constraint for which an impact law is defined. The impact process is considered to be instantaneous and the behavior at the impact is described through a restitution coefficient e. We solve this problem using an algorithm developed and used by Paoli and Schatzman. We show that this algorithm develops numerical instabilities related to the restitution coefficient used in the impact law. Finally, we propose a new scheme which introduces some numerical damping in order to avoid the numerical instabilities.     

A multilevel algorithm for the biharmonic problem

Summary A finite element discretization of the mixed variable formulation of the biharmonic problem is considered. A multilevel algorithm for the numerical solution of the discrete equations is described. Convergence is proved under the assumption ofH ³-regularity.     

Space‐time spectral method for two‐dimensional semilinear parabolic equations

In this paper, a high‐order accurate numerical method for two‐dimensional semilinear parabolic equations is presented. We apply a Galerkin–Legendre spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of ordinary differential equations. Our formulation can be made arbitrarily high‐order accurate in both space and time. Optimal a priori error bound is derived in the L²‐norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence property of the method, show our formulation have spectrally accurate in both space and time. John Wiley & Sons, Ltd.     

An analytical and numerical study of the Stefan problem with convection by means of an enthalpy method

In this paper, we consider a theoretical and numerical study of the Stefan problem with convection, described by the Navier–Stokes equations with no‐slip boundary conditions. The mathematical formulation adopted is based on the enthalpy method. The existence of a weak solution is proved in the bidimensional case. The numerical effectiveness of the model considered is confirmed by some numerical results. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion‐acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear GRLW equation by Petrov–Galerkin method in which the element shape functions are cubic and weight functions are quadratic B‐splines. The proposed method is implemented to three reference problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi‐discrete Galerkin scheme of the equation are firstly constituted. We estimate rate of convergence of such an approximation. Using Fourier stability analysis of the linearized scheme we show that the scheme is unconditionally stable. To verify practicality and robustness of the new scheme error norms L₂, L_∞ and three invariants I₁, I₂, and I₃ are calculated. The computed numerical results are compared with other published results and confirmed to be precise and effective.     

A mixed formulation for the direct approximation of <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-weighted controls for the linear heat equation

This paper deals with the numerical computation of null controls for the linear heat equation. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a given positive time. In [Fernandez-Cara & Münch, Strong convergence approximations of null controls for the 1D heat equation, 2013], a so-called primal method is described leading to a strongly convergent approximation of distributed control: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality conditions. In this work, we adapt the method to approximate the control of minimal square integrable-weighted norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner situation and is valid in any dimension.     

Recovering Mesh Geometry from a Stiffness Matrix

We introduce the following class of mesh recovery problems: Given a stiffness matrix A and a PDE, construct a mesh M such that the finite-element formulation of the PDE over M is A. We show, under certain assumptions, that it is possible to reconstruct the original mesh for the special case of the Laplace operator discretized on an unstructured mesh of triangular elements with linear basis functions. The reconstruction is achieved through a series of techniques from graph theory and numerical analysis, some of which are new and can find application in other scientific areas. Finally, we discuss extensions to other operators and some open questions related to this class of problems.