Robin Conditions Research Articles

An exact solution to a Stefan problem with variable thermal conductivity and a Robin boundary condition

In this article it is proved the existence of similarity solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity and a Robin condition at the fixed face. The temperature distribution is obtained through a generalized modified error function which is defined as the solution to a nonlinear ordinary differential problem of second order. It is proved that the latter has a unique non-negative bounded analytic solution when the parameter on which it depends assumes small positive values. Moreover, it is shown that the generalized modified error function is concave and increasing, and explicit approximations are proposed for it. Relation between the Stefan problem considered in this article with those with either constant thermal conductivity or a temperature boundary condition is also analysed.

Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L1 data

In the present paper, we study the homogenization of a class of quasilinear elliptic problems in a periodically perforated domain Ωε, with L1 data and nonlinear Robin conditions on the boundary of the holes. Considering that we deal with L1 data, we cannot have solutions in H1(Ωε). Therefore, we use here the convenient notion of renormalized solutions. For the homogenization, we use the periodic unfolding method but we can only apply it to the truncated solutions, which are in H1(Ωε). Hence, as a main difficulty, we have to carefully describe the limits of the truncated unfolded solutions and of their gradients. This allow us to prove that we obtain at the limit an unfolded renormalized problem, as well as a (renormalized) homogenized problem in Ω.

A non-invasive implementation of a mixed domain decomposition method for frictional contact problems

A non-invasive implementation of the Latin domain decomposition method for frictional contact problems is described. The formulation implies to deal with mixed (Robin) conditions on the faces of the subdomains, which is not a classical feature of commercial software. Therefore we propose a new implementation of the linear stage of the Latin method with a non-local search direction built as the stiffness of a layer of elements on the interfaces. This choice enables us to implement the method within the open source software Code_Aster, and to derive 2D and 3D examples with similar performance as the standard Latin method.

Comparison of Solution Options for Line-Source Generated Short-Waves in a Wedge with Neumann and Robin Conditions on Respective Faces: Application to Waves on a Plane Beach

High-frequency waves generated by a moving oscillatory line source over a plane beach are examined in the context of a linear non-hydrostatic theory. The work complements a recent study by the author of low-frequency wave generation in a similar environment (Q. Jl. Mech. Appl. Math.68 (2015) 421–460) and provides a special case of a wider class of problems to which belong also a number of problems on electromagnetic diffraction by impedance wedges currently of considerable interest. The latter are generally treatable only by numerical techniques and the opportunity is seized here to examine a comparison between such numerically generated solutions and analytic solutions which can be and are derived and evaluated for the water wave problem. In common with the electromagnetic problems, the water-wave solution is formulated in terms of an inverse Kontorovich–Lebedev transform the inversion of which is considered both by direct quadrature and by residue composition while the spectral function of the transform is also determined in two different ways namely as (i) (for special wedge angles) a semi-closed form entire function and (ii) a numerical solution of a singular integral equation (as commonly encountered in the electromagnetic problems) valid for more general wedge angles. It is concluded that the residue composition technique can be fraught with difficulties and, where spectral function structure allows, should be replaced by the strategy of direct numerical inversion. For the water-wave problem, the anticipated resonances at edge wave frequencies are shown to exist and their strengths prove to be consistent with the well-known findings of Ursell though the process of successfully detecting these purely from the numerical model is shown to be somewhat error prone.

A stable partitioned FSI algorithm for rigid bodies and incompressible flow. Part II: General formulation

A stable partitioned algorithm is developed for fluid–structure interaction (FSI) problems involving viscous incompressible flow and rigid bodies. This added-mass partitioned (AMP) algorithm remains stable, without sub-iterations, for light and even zero mass rigid bodies when added-mass and viscous added-damping effects are large. The scheme is based on a generalized Robin interface condition for the fluid pressure that includes terms involving the linear acceleration and angular acceleration of the rigid body. Added mass effects are handled in the Robin condition by inclusion of a boundary integral term that depends on the pressure. Added-damping effects due to the viscous shear forces on the body are treated by inclusion of added-damping tensors that are derived through a linearization of the integrals defining the force and torque. Added-damping effects may be important at low Reynolds number, or, for example, in the case of a rotating cylinder or rotating sphere when the rotational moments of inertia are small. In this second part of a two-part series, the general formulation of the AMP scheme is presented including the form of the AMP interface conditions and added-damping tensors for general geometries. A fully second-order accurate implementation of the AMP scheme is developed in two dimensions based on a fractional-step method for the incompressible Navier–Stokes equations using finite difference methods and overlapping grids to handle the moving geometry. The numerical scheme is verified on a number of difficult benchmark problems.

Degenerate scale for 2D Laplace equation with Robin boundary condition

It is well known that the 2D Laplace Dirichlet boundary value problem with a specific contour has a degenerate scale for which the boundary integral equation (BIE) has several solutions. We study here the case of the Robin condition (i.e. convection condition for thermal conduction problems), and show that this problem has also one degenerate scale. The cases of the interior problem and of the exterior problem are quite different. For the Robin interior problem, the degenerate scale is the same as for the Dirichlet problem. For the Robin exterior problem, the degenerate scale is always larger than for the Dirichlet problem and has some asymptotic properties. The cases of several simple boundaries like ellipse, equilateral triangle, square and rectangle are numerically investigated and the results are compared with the analytically predicted asymptotic behavior.An important result is that avoiding a contour leading to a degenerate Robin problem cannot be achieved as simply as in the case of Dirichlet boundary condition by introducing a large reference scale into the Green's function.

A stable and accurate partitioned algorithm for conjugate heat transfer

We describe a new partitioned approach for solving conjugate heat transfer (CHT) problems where the governing temperature equations in different material domains are time-stepped in an implicit manner, but where the interface coupling is explicit. The new approach, called the CHAMP scheme (Conjugate Heat transfer Advanced Multi-domain Partitioned), is based on a discretization of the interface coupling conditions using a generalized Robin (mixed) condition. The weights in the Robin condition are determined from the optimization of a condition derived from a local stability analysis of the coupling scheme. The interface treatment combines ideas from optimized-Schwarz methods for domain-decomposition problems together with the interface jump conditions and additional compatibility jump conditions derived from the governing equations. For many problems (i.e. for a wide range of material properties, grid-spacings and time-steps) the CHAMP algorithm is stable and second-order accurate using no sub-time-step iterations (i.e. a single implicit solve of the temperature equation in each domain). In extreme cases (e.g. very fine grids with very large time-steps) it may be necessary to perform one or more sub-iterations. Each sub-iteration generally increases the range of stability substantially and thus one sub-iteration is likely sufficient for the vast majority of practical problems.The CHAMP algorithm is developed first for a model problem and analyzed using normal-mode theory. The theory provides a mechanism for choosing optimal parameters in the mixed interface condition. A comparison is made to the classical Dirichlet–Neumann (DN) method and, where applicable, to the optimized-Schwarz (OS) domain-decomposition method. For problems with different thermal conductivities and diffusivities, the CHAMP algorithm outperforms the DN scheme. For domain-decomposition problems with uniform conductivities and diffusivities, the CHAMP algorithm performs better than the typical OS scheme with one grid-cell overlap. The CHAMP scheme is also developed for general curvilinear grids and CHT examples are presented using composite overset grids that confirm the theory and demonstrate the effectiveness of the approach.

Similarity solution for a two-phase one-dimensional Stefan problem with a convective boundary condition and a mushy zone model

A two-phase solidification process for a one-dimensional semi-infinite material is considered. It is assumed that it is ensued from a constant bulk temperature present in the vicinity of the fixed boundary, which it is modelled through a convective condition (Robin condition). The interface between the two phases is idealized as a mushy region and it is represented following the model of Solomon, Wilson, and Alexiades. An exact similarity solution is obtained when a restriction on data is verified, and it is analysed the relation between the problem considered here and the problem with a temperature condition at the fixed boundary. Moreover, it is proved that the solution to the problem with the convective boundary condition converges to the solution to a problem with a temperature condition when the heat transfer coefficient at the fixed boundary goes to infinity, and it is given an estimation of the difference between these two solutions. Results in this article complete and improve the ones obtained in Tarzia (Comput Appl Math 9:201–211, 1990).

Superconvergent second order Cartesian method for solving free boundary problem for invadopodia formation

In this paper, we present a superconvergent second order Cartesian method to solve a free boundary problem with two harmonic phases coupled through the moving interface. The model recently proposed by the authors and colleagues describes the formation of cell protrusions. The moving interface is described by a level set function and is advected at the velocity given by the gradient of the inner phase. The finite differences method proposed in this paper consists of a new stabilized ghost fluid method and second order discretizations for the Laplace operator with the boundary conditions (Dirichlet, Neumann or Robin conditions). Interestingly, the method to solve the harmonic subproblems is superconvergent on two levels, in the sense that the first and second order derivatives of the numerical solutions are obtained with the second order of accuracy, similarly to the solution itself. We exhibit numerical criteria on the data accuracy to get such properties and numerical simulations corroborate these criteria. In addition to these properties, we propose an appropriate extension of the velocity of the level-set to avoid any loss of consistency, and to obtain the second order of accuracy of the complete free boundary problem. Interestingly, we highlight the transmission of the superconvergent properties for the static subproblems and their preservation by the dynamical scheme. Our method is also well suited for quasistatic Hele–Shaw-like or Muskat-like problems.

Steklov approximations of harmonic boundary value problems on planar regions

Error estimates for approximations of solutions of Laplace’s equation with Dirichlet, Robin or Neumann boundary value conditions are described. The solutions are represented by orthogonal series using the harmonic Steklov eigenfunctions. Error bounds for partial sums involving the lowest eigenfunctions are found. When the region is a rectangle, explicit formulae for the Steklov eigenfunctions and eigenvalues are known. These were used to find approximations for problems with known explicit solutions. Results about the accuracy of these solutions, as a function of the number of eigenfunctions used, are given.

A stable partitioned FSI algorithm for rigid bodies and incompressible flow. Part I: Model problem analysis

A stable partitioned algorithm is developed for fluid–structure interaction (FSI) problems involving viscous incompressible flow and rigid bodies. This added-mass partitioned (AMP) algorithm remains stable, without sub-iterations, for light and even zero mass rigid bodies when added-mass and viscous added-damping effects are large. The scheme is based on a generalized Robin interface condition for the fluid pressure that includes terms involving the linear acceleration and angular acceleration of the rigid body. Added-mass effects are handled in the Robin condition by inclusion of a boundary integral term that depends on the pressure. Added-damping effects due to the viscous shear forces on the body are treated by inclusion of added-damping tensors that are derived through a linearization of the integrals defining the force and torque. Added-damping effects may be important at low Reynolds number, or, for example, in the case of a rotating cylinder or rotating sphere when the rotational moments of inertia are small. In this first part of a two-part series, the properties of the AMP scheme are motivated and evaluated through the development and analysis of some model problems. The analysis shows when and why the traditional partitioned scheme becomes unstable due to either added-mass or added-damping effects. The analysis also identifies the proper form of the added-damping which depends on the discrete time-step and the grid-spacing normal to the rigid body. The results of the analysis are confirmed with numerical simulations that also demonstrate a second-order accurate implementation of the AMP scheme.

Exponential decay and blow-up results for a nonlinear heat equation with a viscoelastic term and Robin conditions

Exponential decay and blow-up results for a nonlinear heat equation with a viscoelastic term and Robin conditions

Trefftz method in solving Fourier-Kirchhoff equation for two-phase flow boiling in a vertical rectangular minichannel

This paper presents the results of investigations into flow boiling heat transfer in an asymmetrically heated vertical minichannel of 1.7 mm depth. The heated element for FC-72 flowing in the minichannel was an alloy plate 0.45 mm thick, microstructured on one side, in direct contact with the flowing fluid. The computational part of the study contains approximate steady state solutions of the heat transfer problems described by Poisson.s equation and the energy equation for the heated plate and the fluid, respectively. For both equations, the boundary conditions were specified on the basis of experimental data. Temperature of the outer plate surface, measured by infrared thermography, and heat losses to ambient air were included in the calculations. For the energy equation we assumed parabolic profile of fluid velocity and the equality of temperatures and heat fluxes at the interface between the heated surface and the fluid. The void fraction was taken from a single-phase flow model. Two-dimensional temperature distributions were obtained by the Trefftz method and, due to the Robin condition at the interface between them, it was possible to calculate the heat transfer coefficient. Its values were compared to those obtained by other correlations known from literature.

Domain decomposition algorithms for the two dimensional nonlinear Schrödinger equation and simulation of Bose–Einstein condensates

In this paper, we apply the optimized Schwarz method to the two dimensional nonlinear Schrodinger equation and extend this method to the simulation of Bose-Einstein condensates (Gross-Pitaevskii equation). We propose an extended version of the Schwartz method by introducing a preconditioned algorithm. The two algorithms are studied numerically. The experiments show that the preconditioned algorithm improves the convergence rate and reduces the computation time. In addition, the classical Robin condition and a newly constructed absorbing condition are used as transmission conditions.

Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions

In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and unfavorable regions for a species to survive. The mathematical formulation of the model leads to an indefinite weight linear eigenvalue problem in a fixed box $$\Omega $$ and we consider the general case of Robin boundary conditions on $$\partial \Omega $$ . It is well known that it suffices to consider bang-bang weights taking two values of different signs, that can be parametrized by the characteristic function of the subset E of $$\Omega $$ on which resources are located. Therefore, the optimal spatial arrangement is obtained by minimizing the positive principal eigenvalue with respect to E, under a volume constraint. By using symmetrization techniques, as well as necessary optimality conditions, we prove new qualitative results on the solutions. Namely, we completely solve the problem in dimension 1, we prove the counter-intuitive result that the ball is almost never a solution in dimension 2 or higher, despite what suggest the numerical simulations. We also introduce a new rearrangement in the ball allowing to get a better candidate than the ball for optimality when Neumann boundary conditions are imposed. We also provide numerical illustrations of our results and of the optimal configurations.

An analytical approach for the simulation of flow in a heterogeneous confined aquifer with a parameter zonation structure

AbstractThis study introduces an analytical approach to estimate drawdown induced by well extraction in a heterogeneous confined aquifer with an irregular outer boundary. The aquifer domain is divided into a number of zones according to the zonation method for representing the spatial distribution of a hydraulic parameter field. The lateral boundary of the aquifer can be considered under the Dirichlet, Neumann or Robin condition at different parts of the boundary. Flow across the interface between two zones satisfies the continuities of drawdown and flux. Source points, each of which has an unknown volumetric rate representing the boundary effect on the drawdown, are allocated around the boundary of each zone. The solution of drawdown in each zone is expressed as a series in terms of the Theis equation with unknown volumetric rates from the source points. The rates are then determined based on the aquifer boundary conditions and the continuity requirements. The estimated aquifer drawdown by the present approach agrees well with a finite element solution developed based on the Mathematica function NDSolve. As compared with the existing numerical approaches, the present approach has a merit of directly computing the drawdown at any given location and time and therefore takes much less computing time to obtain the required results in engineering applications.

Homogenization of elliptic problems with quadratic growth and nonhomogenous Robin conditions in perforated domains

This paper deals with the homogenization of a class of nonlinear elliptic problems with quadratic growth in a periodically perforated domain. The authors prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. The main difficulty, when passing to the limit, is that the solution of the problems converges neither strongly in L2(Ω) nor almost everywhere in Ω. A new convergence result involving nonlinear functions provides suitable weak convergence results which permit passing to the limit without using any extension operator. Consequently, using a corrector result proved in [Chourabi, I. and Donato, P., Homogenization and correctors of a class of elliptic problems in perforated domains, Asymptotic Analysis, 92(1), 2015, 1–43, DOI: 10.3233/ASY-151288], the authors describe the limit problem, presenting a limit nonlinearity which is different for the two cases, that of a Neumann datum with a nonzero average and with a zero average.

On a nonlinear heat equation with viscoelastic term associated with Robin conditions

This paper is devoted to study of a nonlinear heat equation with a viscoelastic term associated with Robin conditions. At first, by the Faedo–Galerkin and the compactness method, we prove existence, uniqueness, and regularity of a weak solution. Next, we prove that any weak solution with negative initial energy will blow up in finite time. Finally, by the construction of a suitable Lyapunov functional, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions.

Existence and uniqueness results for a class of singular elliptic problems in perforated domains

In this paper we prove some existence, uniqueness and regularity results for a weak solution of a quasilinear elliptic nonlinear problem posed in a domain perforated by one or more holes. The nonlinear term is of the form \(f\zeta (u)\), where \(\zeta (s)\) is singular at \(s=0\). On the boundary of the holes we impose a nonlinear Robin condition, while on the exterior boundary we prescribe a homogeneous Dirichlet condition. The difficulty here arises in dealing simultaneously with the quasilinear matrix field, the singular datum and the nonlinear Robin condition. To show the existence of a solution we approximate the problem with a sequence of nonsingular problems for which the existence of a solution is proved via the Schauder fixed-point theorem. The main tool when passing to the limit in the approximate problem is to split the integral of the singular term into the sum of two integrals, one on the set where the solution is very close to the singularity and one where it is far from it. To obtain the uniqueness of the solution we need to require additional assumptions on the quasilinear term and a monotonicity property for the singular one. Under suitable stronger hypotheses on the data, we also prove the boundedness of the solution.

Diamagnetism versus Robin condition and concentration of ground states

We estimate the ground state energy for the magnetic Laplacian with a Robin condition. In a special asymptotic limit, we find that the magnetic field does not contribute to the two-term expansion of the ground state energy, thereby proving that the Robin condition weakens diamagnetism. We discuss a semi-classical version of the operator and prove that the ground states concentrate near the boundary points of maximal curvature.