Identification of a free boundary in Norton–Hoff flows with thermal effects

This work deals with a free boundary identification problem in a steady viscoplastic flow. We provide a novel identification model based on a non-linear optimization. The fluid motion is governed by the incompressible Norton–Hoff model coupled with the heat equation. The viscosity of the fluid is modeled by the non-linear Arrhenius law. Our point of view is to treat the problem as a shape sensitivity of a cost functional formulated on the free boundary and governed by the normal component of the velocity of the flow. We analyze the mathematical statement of the forward problem. The equations related to the free boundary are simplified. Various properties of this optimization are proved. Since the state of Norton–Hoff model is not regular enough we introduce a parameter penalization. The shape gradient of the considered cost functional is given in the strong sense up to the parameter of penalization. We supply the expression of the shape gradient in a weak sense.     

Shape optimization of a body in the Oseen flow

This article is concerned with a numerical simulation of shape optimization of the Oseen flow around a solid body. The shape gradient for shape optimization problem in a viscous incompressible flow is computed by the velocity method. The flow is governed by the Oseen equations with mixed boundary conditions containing the pressure. The structure of continuous shape gradient of the cost functional is derived by using the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. A gradient type algorithm is applied to the shape optimization problem. Numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Study of a Numerical Approach for a Transient Flow Problem in Porous Media

In this work, we deal with the numerical study of the new approximation method proposed in [7] for a transient flow problem in porous media. The stationary problem, obtained from a time discretization of this transient problem, is considered as an optimal shape design formulation. We prove the existence of the solution of the discrete optimal shape problem obtained from finite element discretization. We study the convergence and give numerical results showing the efficiency of the proposed approach.     

Control of the jet of a pulsed flywheel water cannon

We describe a pulsed water cannon in which the compression of the water occurs as a result of a cam located on a flywheel. We pose and solve the variational problem of optimal control of the spray of the pulsed water cannon. The control is the shape of the cam. We compute a way of finding a pressure pulse that is nearly rectangular, providing a uniform fluid flow. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 107–111.     

The steady two-dimensional flow over a rectangular obstacle lying on the bottom

We study a plane problem with mixed boundary conditions for a harmonic function in an unbounded Lipschitz domain contained in a strip. The problem is obtained by linearizing the hydrodynamic equations which describe the steady flow of a heavy ideal fluid over an obstacle lying on the flat bottom of a channel. In the case of obstacles of rectangular shape we prove unique solvability for all velocities of the (unperturbed) flow above a critical value depending on the obstacle depth. We also discuss regularity and asymptotic properties of the solutions.     

Solving a free boundary problem with nonconstant coefficients

The present article is concerned with the numerical solution of a free boundary problem for an elliptic state equation with nonconstant coefficients. We maximize the Dirichlet energy functional over all domains of fixed volume. The domain under consideration is represented by a level set function, which is driven by the objective's shape gradient. The state is computed by the finite element method where the underlying triangulation is constructed by means of a marching cubes algorithm. We show that the combination of these tools lead to an efficient solver for general shape optimization problems.     

On the inclusion of a map into a flow

We consider the problem of the inclusion of a diffeomorphism into a flow generated by an autonomous or time periodic vector field. We discuss various aspects of the problem, present a series of results (both known and new ones) and point out some unsolved problems.     

On Optimal Control of a Free Surface Flow

We discuss an optimal control approach for a 2D Stokes flow with a free surface. The aim is to optimize the shape of a polymer film by adjusting the ambient pressure in a casting process. The resulting minimization problem is solved by the method of steepest descent. Numerical results will be presented. Furthermore we state the adjoint system for the Lagrangian formalism. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Slow viscous flow near a superhydrophobic surface with trapped gas bubbles

The Boundary Element Method (BEM) is used to solve the problem of Stokes flow of a viscous fluid over a periodic striped texture of a superhydrophobic surface (SHS), partially filled with frictionless gas bubbles. The shape of the bubble surfaces and the position of the meniscus pinning points relative to the cavity walls are taken into account in the study. Two kinds of flows important for practical applications are considered: a pressure-driven flow in a thin channel with a bottom superhydrophobic wall and a shear-driven flow over a periodic texture. We study the flow pattern in the fluid over a single cavity containing a bubble with a curved phase interface shifted into the cavity. A parametric numerical study of the averaged slip length of the SHS is performed as a function of the geometric parameters of the texture. It is shown that the curvature of the phase interface and/or its shift into the cavity both result in the decrease in the average slip length. It is demonstrated that the BEM can be an efficient tool for studying Stokes flows over textured superhydrophobic surfaces with different geometries of microcavities and phase interfaces. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuous Frechet differentiability with respect to a Lipschitz domain and a stability estimate for direct acoustic scattering problems

We consider direct acoustic scattering problems with eithera sound-soft or sound-hard obstacle, or lossy boundary conditions,and establish continuous Fréchet differentiability withrespect to the shape of the scatterer of the scattered fieldand its corresponding far-field pattern. Our proof is basedon the implicit function theorem, and assumes that the boundaryof the scatterer as well as the deformation are only Lipschitzcontinuous. From continuous Fréchet differentiability,we deduce a stability estimate governing the variation of thefar-field pattern with respect to the shape of the scatterer.We illustrate this estimate with numerical results obtainedfor a two-dimensional high-frequency acoustic scattering problem.     

An optimal shape design formulation for inhomogeneous dam problems

In this paper, the flow problem of incompressible liquid through an inhomogeneous porous medium (say dam), with permeability allowing parametrization of the free boundary by a graph of continuous unidimensional function, is considered. We propose a new formulation on an optimal shape design problem. We show the existence of a solution of the optimal shape design problem. The finite element method is used to obtain numerical results which show the efficiency of the proposed approach. Copyright © 2002 John Wiley & Sons, Ltd.     

Shape inverse problem for the two‐dimensional unsteady Stokes flow

In this article, the shape inverse problem for the two‐dimensional unsteady Stokes flow has been presented. We employ Piola transformation to bypass the divergence free condition for the flow and prove the differentiability of the solution to the initial boundary value problem. For the approximate solution of the ill‐posed and nonlinear problem, we propose a regularized Gauss‐Newton method. The numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

On the Existence and Stability of a Global Subsonic Flow in a 3D Infinitely Long Cylindrical Nozzle

This paper is concerned with the problem on the global existence and stability of a subsonic flow in an infinitely long cylindrical nozzle for the 3D steady potential flow equation. Such a problem was indicated by Courant-Friedrichs in [8, p. 377]： A flow through a duct should be considered as a cal symmetry and should be determined steady, isentropic, irrotational flow with cylindriby solving the 3D potential flow equations with appropriate boundary conditions. By introducing some suitably weighted HSlder spaces and establishing a priori estimates, the authors prove the global existence and stability of a subsonic potential flow in a 3D nozzle when the state of subsonic flow at negative infinity is given.     

The Asymptotic Behaviour of a Curved Line Source Plume within an Enclosure

A line source of heat is situated in an enclosure with inletsand an outlet for the contained fluid. The convection plumeformed must take a curved shape because of the asymmetry ofthe enclosure. We show how boundary-layer theory can be usedto find this shape in the case where the plume's deflectionis small. Two effects govern the shape of the curve; a buoyancyeffect and an inertial effect due to the bounding walls. Bysolving the outer potential flow problem, both effects can beincluded in the model. Finally, a stability analysis is carriedout for steady plumes which shows that any steady solution withan inflection point, as well as several other solutions, isunstable. A comparison is made with previous work which doesnot include all these effects, and with some experimental results.     

The Alexander transform of a spirallike function

We show that the Alexander transform of a β-spirallike function is univalent when cosβ?1/2, which settles the problem posed by Robertson. We also solved a problem considered by Y.J. Kim and Merkes.     

A Fast Iterative Method to Compute the Flow Around a Submerged Body

We develop an efficient iterative method for computing the steady linearized potential flow around a submerged body moving in a liquid of finite constant depth. In this paper we restrict the presentation to the two-dimensional problem, but the method is readily generalizable to the three-dimensional case, i.e., the flow in a canal. The problem is indefinite, which makes the convergence of most iterative methods unstable. To circumvent this difficulty, we decompose the problem into two more easily solvable subproblems and form a Schwarz--type iteration to solve the original problem. The first subproblem is definite and can therefore be solved by standard iterative methods. The second subproblem is indefinite but has no body. It is therefore easily and efficiently solvable by separation of variables. We prove that the iteration converges for sufficiently small Froude numbers. In addition, we present numerical results for a second-order accurate discretization of the problem. We demonstrate that the iterative method converges rapidly, and that the convergence rate improves when the Froude number decreases. We also verify numerically that the convergence rate is essentially independent of the grid size.

     

用正则化方法求解声波散射反问题

研究了从声波散射场的远场模式的信息来再现散射物边界形状的反问题.首先构造表达散射物特征的指示函数,然后利用该函数之特性,建立求解该类反问题的基本方程,从而确定散射物的边界形状.在这个算法中,不需预先知道散射物的边界类型和形状等知识,从T ikhonov正则化方法进行的数值计算结果表明了该方法是有效的和实用的.     

On a Kohn-Vogelius like formulation of free boundary problems

The present paper is concerned with the solution of a Bernoulli type free boundary problem by means of shape optimization. Two state functions are introduced, namely one which satisfies the mixed boundary value problem, whereas the second one satisfies the pure Dirichlet problem. The shape problem under consideration is the minimization of the L ²-distance of the gradients of the state functions. We compute the corresponding shape gradient and Hessian. By the investigation of sufficient second order conditions we prove algebraic ill-posedness of the present formulation. Our theoretical findings are supported by numerical experiments.     

Solving inverse boundary-value problem of aerohydrodynamics in a new statement

We consider a modified inverse boundary-value problem of aerohydrodynamics, in which it is required to find the shape of a wing profile, streamlined by a potential flow of incompressible inviscid fluid, when the distribution of the velocity potential on one section of the profile is given as a function of abscissa, and on the rest of the profile as a function of the ordinate of the profile point.