A three field stabilized finite element method for the Stokes equationsUne méthode d'éléments finis stabilisée à trois champs pour les équations de Stokes

We consider in this work the boundary value problem for Stokes equations on a two dimensional domain in cases where non-standard boundary conditions are given. We study the cases where pressure and normal or tangential components of the velocity are given in different parts of the boundary and solve the problem with a minimal regularity. We introduce the problem and its variational formulation which is a mixed one. The principal unknowns are the pressure and the vorticity, the multiplier is the velocity. We present the numerical discretization which needs some stabilization. We prove the convergence and the behavior of the a priori error estimates. Some numerical tests are also presented. To cite this article: M. Amara et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 603–608.     

Algorithms for vector field generation in mass consistent models

Diagnostic models in meteorology are based on the fulfillment of some time independent physical constraints as, for instance, mass conservation. A successful method to generate an adjusted wind field, based on mass conservation equation, was proposed by Sasaki and leads to the solution of an elliptic problem for the multiplier. Here we study the problem of generating an adjusted wind field from given horizontal initial velocity data, by two ways. The first one is based on orthogonal projection in Hilbert spaces and leads to the same elliptic problem but with natural boundary conditions for the multiplier. We derive from this approach the so called E–algorithm. An innovative alternative proposal is obtained from a second approach where we consider the saddle–point formulation of the problem—avoiding boundary conditions for the multiplier— and solving this problem by iterative conjugate gradient methods. This leads to an algorithm that we call the CG–algorithm, which is inspired from Glowinsk's approach to solve Stokes–like problems in computational fluid dynamics. Finally, the introduction of new boundary conditions for the multiplier in the elliptic problem generates better adjusted fields than those obtained with the original boundary conditions. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Turbulence modeling from a new perspective

In this work we establish a link between a Reynolds averaged turbulence modeling methodology, containing interactions up to the second order correlations between the velocity fluctuations at various scales, and a multi-objective optimization problem with the constraints expressed in terms of equality and inequality, imposed by the given boundary conditions and the positive semi-definiteness of the Reynolds stress tensor, etc. The information unavailability and uncertainty associated with the boundary conditions for the fluctuation correlations of various orders is delineated, and the information from the Navier–Stokes equations is utilized to the extent allowed by the available input data necessary for simulations; turbulence from the perspective of systems simulation is explored and some objective functions are proposed. Finally, the challenges faced by the formulation and the issues yet to be resolved are discussed.     

Error estimates for three-dimensional Stokes problem with non-standard boundary conditions

This work is devoted to the optimal and a posteriori error estimates of the Stokes problem with some non-standard boundary conditions in three dimensions. The variational formulation is decoupled into a system for the velocity and a Poisson equation for the pressure. The velocity is approximated with curl conforming finite elements and the pressure with standard continuous elements. Next, we establish optimal a posteriori estimates.     

Schwarz alternating domain decomposition approach for the solution of two‐dimensional Navier–Stokes flow problems by the method of approximate particular solutions

The method of approximate particular solutions (MAPS) is used to solve the two‐dimensional Navier–Stokes equations. This method uses particular solutions of a nonhomogeneous Stokes problem, with the multiquadric radial basis function as a nonhomogeneous term, to approximate the velocity and pressure fields. The continuity equation is not explicitly imposed since the used particular solutions are mass conservative. To improve the computational efficiency of the global MAPS, the domain is split into overlapped subdomains where the Schwarz Alternating Algorithm is employed using velocity or traction values from neighboring subdomains as boundary conditions. When imposing only velocity boundary conditions, an extra step is required to find a reference value for the pressure at each subdomain to guarantee continuity of pressure across subdomains. The Stokes lid‐driven cavity flow problem is solved to assess the performance of the Schwarz algorithm in comparison to a finite‐difference‐type localized MAPS. The Kovasznay flow problem is used to validate the proposed numerical scheme. Despite the use of relative coarse nodal distributions, numerical results show excellent agreement with respect to results reported in literature when solving the lid‐driven cavity (up to Re = 10,000) and the backward facing step (at Re = 800) problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 777–797, 2015     

A CQM-based BEM for transient heat conduction problems in homogeneous materials and FGMs

A method for numerical solution of time-domain boundary integral formulations of transient problems governed by the heat equation is presented. The heat conduction problem is analyzed considering homogeneous and non-homogeneous media. In the case of the non-homogeneous media, the conductor material is assumed to be a functionally graded material, i.e., the material properties vary spatially according to known smooth functions. For some specific spatial variations of the material properties, the fundamental solution and the boundary integral equation of the problem are obtained thanks to a change of variables that transforms the original problem to the standard heat conduction problem for homogeneous materials. For the treatment of time-dependent terms, the convolution quadrature method is adopted to approximate numerically the integral equation of the time-domain boundary element method. In the case that the responses are required at a large number of interior points, the convolution performed to calculate them is very time consuming. It is shown that the discrete convolution of the proposed formulation can be computed by means of the fast Fourier transform technique, which considerably reduces the computational complexity. Results for some transient heat conduction examples are presented to validate the numerical techniques studied.     

Error analysis of a weighted least-squares finite element method for 2-D incompressible flows in velocity–stress–pressure formulation

In this paper we are concerned with a weighted least-squares finite element method for approximating the solution of boundary value problems for 2-D viscous incompressible flows. We consider the generalized Stokes equations with velocity boundary conditions. Introducing the auxiliary variables (stresses) of the velocity gradients and combining the divergence free condition with some compatibility conditions, we can recast the original second-order problem as a Petrovski-type first-order elliptic system (called velocity–stress–pressure formulation) in six equations and six unknowns together with Riemann–Hilbert-type boundary conditions. A weighted least-squares finite element method is proposed for solving this extended first-order problem. The finite element approximations are defined to be the minimizers of a weighted least-squares functional over the finite element subspaces of the H¹ product space. With many advantageous features, the analysis also shows that, under suitable assumptions, the method achieves optimal order of convergence both in the L²-norm and in the H¹-norm. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

The Stokes equations with Fourier boundary conditions on a wall with asperities

We study the effect of the rugosity of a wall on the solution of the Stokes system complemented with Fourier boundary conditions. We consider the case of small periodic asperities of size ε. We prove that the velocity field, pressure and drag, respectively, converge to the velocity field, pressure and drag of a homogenized Stokes problem, where a different friction coefficient appears. This shows that, contrarily to the case of Dirichlet boundary conditions, rugosity is dominant here. Copyright © 2001 John Wiley & Sons, Ltd.     

A computational determination of the Cowper–Symonds parameters from a single Taylor test

A novel technique for the dynamic characterization of metals from a single Taylor impact test is proposed. This computational characterization procedure is based on the formulation and solution of a first class inverse problem, in which the silhouette of the Taylor specimen’s final shape is expressed as a vector of its geometrical moments and used as input parameter. The inverse characterization problem is reduced to an optimization problem where the optimum material parameters for the Cowper–Symonds material model are determined. The optimization process is performed by a range adaptation real-coded genetic algorithm. Numerical example for the characterisation of 1018 steel is implemented and presented to validate the methodology presented in this paper. The effectiveness and simplicity of the proposed characterization procedure makes it an appropriate tool for the characterization of metals at high strain rates.     

Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity–pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier–Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.     

Adapted kussmaul formulation of acoustic scattering problems: an overview

Problems of exterior acoustic scattering may be conveniently formulated by means of boundary integral equations. The problem seeks to find a wave function which gives velocity potential profile, pressure density profile, etc. of the acoustic wave at points in space. At the background of the formulations are two theories viz. (Helmholtz) Potential theory and the Green's representation formula. Potential theory gives rise to the so-called indirect formulation and the Green's representation formula to the direct formulations. Classical boundary integral formulations fail at the eigenfrequencies of the interior domain. That is, if a solution is sought of the exterior problem by first solving a homogeneous boundary integral equation, one is inevitably led to the conclusion that these homogeneous boundary equations have nontrivial solutions at certain wave-numbers which are the eigenvalues of the corresponding interior problem. At lower wave-numbers, these eigenfrequencies are thinly distributed but the higher the wave-number, the denser it becomes. This is a well-known drawback for both time-harmonic acoustics and elastodynamics. This is not a physical difficulty but arises entirely as a result of a deficiency in the integral equation is representation. Why then use It? The use has many advantages notably in that the meshing region is reduced from the infinite domain exterior to the body to its finite surface. This created the need for some robust formulations. A proof of the Kussmaul [1] formulation is presented. The formulation has a hypersingular kernel in the integral operator, which creates a havoc in computation (e.g., ill conditioning). The hyper-singularity can be avoided [2], as a result a new formulation is proposed. This paper presents a broad overview of the Adapted Kussmaul Formulation (AKF).     

Simplification through regression analysis on the dynamic response of plates with arbitrary boundary conditions excited by moving inertia load

Dynamic response of a thin rectangular plate traversed by a moving inertia load with arbitrary boundary condition is investigated through this paper. The inertia effect of mass is considered and relevant formulation is established based on the full-term of acceleration, employing the method of Boundary Characteristic Orthogonal Polynomials, BCOP. To acquire the complete solution of partial differential equations governing on the plate, the Galerkin method is used to separate the temporal function from the spatial one. The problem is formulated in the state space and applying the numerical method of Matrix Exponential the complete solution would be achieved. In the numerical studies, a comprehensive parametric study is performed for both cases of loading when inertia effect is included or neglected. Several mass and aspect ratios for the plate with major types of boundary conditions CCCC, SSSS, CFCF and SFSF are accounted for presenting the results. Dynamic amplification factor against velocity parameter is scrutinized within many graphs alongside with a time history analysis of dynamic deflection for the plate's mid-span. Investigating on the dynamic response concludes to the critical boundary condition upon moving mass. By introducing a conversion factor, the margin of inertia and the critical velocity where happened would be achieved, then through a regression analysis a curve fitting model of polynomials is proposed. Corresponding coefficients testify the goodness of fit for such regression which are reported within tables. Referring to this simplified model of conversion factor pertaining to the specific boundary condition, it would be possible to handle the problem in moving load case without undertaking the complexities arisen from inertia contribution into the formulation. Having derived the factor from simplified model which has been calculated for a specific mass and velocity ratio, then multiplying into the moving load response, the complete solution for moving mass would be achieved.     

A priori and a posteriori estimates for three‐dimensional Stokes equations with nonstandard boundary conditions

In this article, we study the Stokes problem with some nonstandard boundary conditions. The variational formulation decouples into a system for the velocity and a Poisson equation for the pressure. The corresponding discrete system do not need an inf‐sup condition. Hence, the velocity is approximated with “ curl ” conforming finite elements and the pressure with standard continuous elements. Next, we establish optimal a priori and a posteriori estimates and we finally concluded with numerical tests. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Laminar mixing layer on the boundary of two flows in the presence of a longitudinal pressure gradient : PMM vol. 38, n 6, 1974, pp. 1133–1136

We consider, in a linear formulation, the problem concerning the laminar mixing layer on the boundary of two flows of an incompressible liquid with a small difference in their Bernoulli constants; we assume the presence of longitudinal pressure gradient. We determine the velocity distribution in the mixing layer, the magnitude of the displacement thickness and the momentum loss thickness. For the case in which there is no longitudinal pressure gradient we calculate the force effect of the one flow on the other.     

Numerical solution of the incompressible Navier-Stokes equations with explicit integration of the pressure term

We propose a formulation of the incompressible Navier-Stokes equations considering a Poisson equation with Neumann boundary conditions for the pressure, and innovative boundary conditions for the velocity. Numerical tests show that the proposed formulation ensures solenoidality of the velocity field. If the initial condition is not divergence-free, exponential decay is observed in time for the error in the fulfillment of the continuity equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constraining inverse stefan design problems

A new formulation possessing stable numerical characteristics is presented for inverse Stefan design processes. In such processes, the goal of the analysis is to design transient boundary conditions which produce the desired interfacial surface motion. This subclass of mildly ill-posed mathematical problems is amenable to the proposed solution methodology. This investigation presents a fixed-front differential formulation from which a weighted residual statement is developed. Orthogonal collocation is used to obtain numerical results illustrating the merit of imposing physical constraints in the mathematical model. These mathematical constraints can be viewed as design specifications and are available to the designer or experimentalist. The proposed methodology is flexible and can be generalized to problems involving continuous casting or crystal growth. Finally, symbolic manipulation is used for augmenting the computational methodology.     

A mathematical programming approach to the computation of the omega invariant of a numerical semigroup

In this paper we present a mathematical programming formulation for the ω-invariant of a numerical semigroup for each of its minimal generators which is an useful index in commutative algebra (in particular in factorization theory) to analyze the primality of the elements in the semigroup. The model consists of solving a problem of optimizing a linear function over the efficient set of a multiobjective linear integer program. We offer a methodology to solve this problem and we provide some computational experiments to show the efficiency of the proposed algorithm.     

Vorticity–velocity–pressure formulation for the Stokes problem

We present a new variational formulation of Stokes problem of fluid mechanics that allows to take into account very general boundary conditions for velocity, tangential vorticity or pressure. This formulation conducts a well posed mathematical problem in a family of particular cases. Copyright © 2002 John Wiley & Sons, Ltd.     

Analysis of the coupling of BEM, FEM and mixed-FEM for a two-dimensional fluid–solid interaction problem

In this paper we consider a fluid–solid interaction problem posed in the plane. We employ a mixed variational formulation in the obstacle, in which the Cauchy stress tensor and the rotation are the only unknowns. This new mixed formulation is coupled, through suitable transmission conditions on the wet interface, with a Helmholtz equation satisfied by the pressure of the fluid in the unbounded domain. We use a traditional primal variational formulation in this part of the domain and incorporate the far field information through boundary integral equations. We approximate the resulting weak formulation by a Galerkin scheme based on PEERS in the solid and on a FEM-BEM approach in the fluid part. We show that our scheme is uniquely solvable and convergent, and then provide optimal error estimates. Finally, we illustrate our analysis with some computational experiments.     

Coupling of mixed finite element and stabilized boundary element methods for a fluid–solid interaction problem in 3D

We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well‐known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well‐posed and propose a conforming Galerkin method based on the lowest‐order Arnold–Falk–Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211–1233, 2014