Decoupled smooth interfaces for spectral-element approximations of parabolic or elliptic type

A method is examined to approximate the interface conditions for Chebyshev polynomial approximations to the solutions of parabolic problems, and a smoothing technique is used to calculate the interface conditions for a domain decomposition method. The methods uses a polynomial of one less degree then the full approximation to calculate the first derivative so that interface values can be calculated by using only the adjacent subdomains. Theoretical results are given for the consistency of the scheme and practical results are presented. Computational results are given for both a fourth-order Runga-Kutta methods and an explicit/implicit scheme. © 1993 John Wiley & Sons, Inc.     

Asymptotic analysis of the differences between the Stokes-Darcy system with different interface conditions and the Stokes-Brinkman system

We consider the coupling of the Stokes and Darcy systems with different choices for the interface conditions. We show that, comparing results with those for the Stokes-Brinkman equations, the solutions of Stokes-Darcy equations with the Beavers-Joseph interface condition in the one-dimensional and quasi-two-dimensional (periodic) cases are more accurate than are those obtained using the Beavers-Joseph-Saffman-Jones interface condition and that both of these are more accurate than solutions obtained using a zero tangential velocity interface condition. The zero tangential velocity interface condition is in turn more accurate than the free-slip interface boundary condition. We also prove that the summation of the quasi-two-dimensional solutions converge so that the conclusions are also valid for the two-dimensional case.     

Using Time Scales to Study Multi-Interval Sturm–Liouville Problems with Interface Conditions

We consider a Sturm–Liouville problem defined on multiple intervals with interface conditions. The existence of a sequence of eigenvalues is established and the zero counts of associated eigenfunctions are determined. Moreover, we reveal the continuous and discontinuous nature of the eigenvalues on the boundary condition. The approach in this paper is different from those in the literature: We transfer the Sturm–Liouville problem with interface conditions to a Sturm–Liouville problem on a time scale without interface conditions and then apply the Sturm–Liouville theory for equations on time scales. In this way, we are able to investigate the problem in a global view. Consequently, our results cover the cases when the potential function in the equation is not strictly greater than zero and when the domain consists of an infinite number of intervals.     

On the well-posed coupling between free fluid and porous viscous flows

We present a well-posed model for the Stokes/Brinkman problem with a family of jump embedded boundary conditions (J.E.B.C.) on an immersed interface with weak regularity assumptions. It arises from a general framework recently proposed for fictitious domain problems. Our model is based on algebraic transmission conditions combining the stress and velocity jumps on the interface $\Sigma$ separating the fluid and porous domains. These conditions are well chosen to get the coercivity of the operator. Then, the general framework allows us to prove new results on the global solvability of some models with physically relevant stress or velocity jump boundary conditions for the momentum transport at a fluid–porous interface. The Stokes/Brinkman problem with Ochoa-Tapia and Whitaker (1995) [9], [10] interface conditions and the Stokes/Darcy problem with Beavers and Joseph (1967) [13] conditions are both proved to be well-posed, by an asymptotic analysis. Up to now, only the Stokes/Darcy problem with Saffman (1971) [15] approximate interface conditions with negligible tangential porous velocity was known to be well-posed.     

A uniformly well-conditioned, unfitted Nitsche method for interface problems

A finite element method for elliptic partial differential equations that allows for discontinuities along an interface not aligned with the mesh is presented. The solution on each side of the interface is separately expanded in standard continuous, piecewise-linear functions, and jump conditions at the interface are weakly enforced using a variant of Nitsche’s method. In our method, the solutions on each side of the interface are extended to the entire domain which results in a fixed number of unknowns independent of the location of the interface. A stabilization procedure is included to ensure well-defined extensions. We prove that the method provides optimal convergence order in the energy and the L ² norms and a condition number of the system matrix that is independent of the position of the interface relative to the mesh. Numerical experiments confirm the theoretical results and demonstrate optimal convergence order also for the pointwise errors.     

An asymptotic analysis of the steady process of saturation of a laminated porous material

The propagation of a steady saturation front in a double-layer porous material, situated between impenetrable walls, is investigated. The closed system of equations and boundary conditions are written on the assumption that the displacing and displaced phases, the viscosities of which differ considerably, are connected, and that the capillary pressure on the interface is constant. The features of the behaviour of the interface in the neighbourhood of the boundary of the layers are investigated in the case when the layer thicknesses differ considerably. When the permeabilities of the layers differ considerably, asymptotic expressions are obtained for the pressure and shape of the interface, and a comparison is made with the results of a numerical solution of the complete problem and with the known asymptotic relations obtained when using a simplified boundary condition at the interface.     

High-Temperature Creep of Fibrous Metal-Matrix Composites under Varying Stresses

The present paper is aimed at testing the hypothesis about the failure of the relatively weak fiber/matrix interface under cyclic loading, which causes an increase in the steady-state creep rate. The hypothesis is tested qualitatively by comparing the creep behavior of composite specimens with various interface strengths under the conditions mentioned (loading-unloading-loading to the original level). The hypothesis is tested semi-quantitatively by estimating the interface strength in relation to the action decreasing the strength. The latter requires the use of a microstructural calculation model. Both the approaches are used in the paper, and the results found support this hypothesis. The experimental data obtained are an additional argument for the necessity of developing metal-matrix composites with a strong interface, which can be a basis for real creep-resistant high-temperature composites.     

Domain decomposition for a finite volume method on non-matching grids

We are interested in a robust and accurate domain decomposition method with Robin interface conditions on non-matching grids using a finite volume discretization. We introduce transmission operators on the non-matching grids and define new interface conditions of Robin type. Under a compatibility assumption, we show the equivalence between Robin interface conditions and Dirichlet–Neumann interface conditions and the well-posedness of the global and local problems. Two error estimates are given in terms of the discrete H¹-norm: one in O(h^1/2) with operators based on piecewise constant functions and the other in O(h) (as in the conforming case) with operators using a linear rebuilding. Numerical results are given. To cite this article: L. Saas et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions

A state-constrained optimal control problem with nonlocal radiation interface conditions arising from the modeling of crystal growth processes is considered. The problem is approximated by a Moreau-Yosida type regularization. Optimality conditions for the regularized problem are derived and the convergence of the regularized problems is shown. In the last part of the paper, some numerical results are presented.     

A Well-Conditioned,Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems

In this paper, we introduce a nonconforming Nitsche's extended finite element method (NXFEM) for elliptic interface problems on unfitted triangulation elements. The solution on each side of the interface is separately expanded in the standard nonconforming piecewise linear polynomials with the edge averages as degrees of freedom. The jump conditions on the interface and the discontinuities on the cut edges (the segment of edges cut by the interface) are weakly enforced by the Nitsche's approach. In the method, the harmonic weighted fluxes are used and the extra stabilization terms on the interface edges and cut edges are added to guarantee the stability and the well conditioning. We prove that the convergence order of the errors in energy and $L^2$ norms are optimal. Moreover, the errors are independent of the position of the interface relative to the mesh and the ratio of the discontinuous coefficients. Furthermore, we prove that the condition number of the system matrix is independent of the interface position. Numerical examples are given to confirm the theoretical results.     

Towards efficient interface conditions for a Schwarz domain decomposition algorithm for an advection equation with biharmonic diffusion

This paper addresses the problem of deriving efficient interface conditions for solving biharmonic diffusion–advection equations using a Schwarz global-in-time domain decomposition algorithm. General interface conditions are proposed, which lead to well-posed problems on each subdomain. The equation is then studied in the simplified 1D case. Exact non-local absorbing boundary conditions are derived, and are approximated by optimized local interface conditions, the efficiency of which is illustrated by numerical experiments.     

A new augmented immersed finite element method without using SVD interpolations

Augmented immersed interface methods have been developed recently for interface problems and problems on irregular domains including CFD applications with free boundaries and moving interfaces. In an augmented method, one or several augmented variables are introduced along the interface or boundary so that one can get efficient discretizations. The augmented variables should be chosen such that the interface or boundary conditions are satisfied. The key to the success of the augmented methods often relies on the interpolation scheme to couple the augmented variables with the governing differential equations through the interface or boundary conditions. This has been done using a least squares interpolation (under-determined) for which the singular value decomposition (SVD) is used to solve for the interpolation coefficients. In this paper, based on properties of the finite element method, a new augmented immersed finite element method (IFEM) that does not need the interpolations is proposed for elliptic interface problems that have a piecewise constant coefficient. Thus the new augmented method is more efficient and simple than the old one that uses interpolations. The method then is extended to Poisson equations on irregular domains with a Dirichlet boundary condition. Numerical experiments with arbitrary interfaces/irregular domains and large jump ratios are provided to demonstrate the accuracy and the efficiency of the new augmented methods. Numerical results also show that the number of GMRES iterations is independent of the mesh size and nearly independent of the jump in the coefficient.     

Surface and internal waves: The two-dimensional problem on forward motion of a body intersecting the interface between two fluids

A linear two-dimensional boundary value problem, that describes steady-state surface and internal waves due to the forward motion of a body in a fluid consisting of two superposed layers with different densities, is considered. The body is fully submerged and intersects the interface between the two layers. Two well-posed formulations of the problem are proposed in which, along with the Laplace equation, boundary conditions, coupling conditions on the interface, and conditions at infinity, a pair of supplementary conditions are imposed at the points where the body contour intersects the interface. In one of the well-posed formulations (where the differences between the horizontal momentum components are given at the intersection points), the existence of the unique solution is proved for all values of the parameters except for a certain (possibly empty) nowhere dense set of values.     

Conditions d'interface pour les méthodes de décomposition de domaine pour le système d'Oseen en dimensions 2 et 3

We study domain decomposition methods for two- and three-dimensional Oseen equations. We propose pseudo-differential interface conditions which lead to the convergence of an additive Schwarz type method in a number of steps equal to the number of subdomains. These conditions are approximated by partial differential conditions for which convergence is proved. These results generalize previous works on scalar equations.     

Interface optimality in fuzzy inference systems

In this paper we address the issue of designing optimal fuzzy interfaces, which are fundamental components of a fuzzy inference system. Due to the different roles of input and output interfaces, optimality conditions are analyzed separately for the two types of interface. We prove that input interfaces are optimal when based on a particular class of fuzzy sets called “bi-monotonic”, provided that mild conditions hold. The class of bi-monotonic fuzzy sets covers a broad range of fuzzy sets shapes, including convex fuzzy sets, so that the provided theoretical results can be applied to several fuzzy models. Such theoretical results are not applicable to output interfaces, for which a different optimality criterion is proposed. Such criterion leads to the definition of an optimality degree that measures the quality of a fuzzy output interface. Illustrative examples are presented to highlight the features of the proposed optimality degree in assessing the quality of output interfaces.     

Mixed boundary value problems and parametrices in the edge calculus

Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. We develop a new method for computing the interface conditions in terms of the index of boundary value problems in weighted spaces on infinite cones, combined with structures from the calculus of boundary value problems on a manifold with edges. This will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator. The approach itself is completely general.     

Steady-state motion of a mode-III crack on imperfect interfaces

The aim of the present paper is to analyse the behaviour ofthe stress and displacement fields in the vicinity of the tipof a crack moving along a bi-material interface. For simplicity,we consider a straight interface of infinite extent. We assumethat the two phases are separated by a thin layer which is either‘soft’ or ‘stiff’ compared to the othertwo phases. We derive the transmission conditions which takeinto account the material properties of the layer and modelthe way the load is transferred across the layer from one phaseto the other. We assume that the point of interchange in theboundary/transmission conditions coincides with the crack tipthat moves along the interface boundary with a constant speed.We develop an integral equation formulation and derive asymptoticformulae for the out-of-plane displacement and the Mode-IIIstress intensity factor associated with such a motion of thecrack inside the interphase layer. The theoretical results areillustrated by numerical examples.     

An interface-fitted adaptive mesh method for elliptic problems and its application in free interface problems with surface tension

This work presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by two operations: (i) the projection of mesh nodes onto the interface and (ii) the insertion of mesh nodes right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, the interface surface tension serves as the jump condition or the Dirichlet boundary condition of the pressure, and the pressure is solved with the interface-fitted finite element method developed in this work. In this study, a level-set function is used to capture the evolution of the interface and provide the interface location for the interface fitting.     

Absorbing Interface Conditions for the Simulation of Wave Propagation on Non-Uniform Meshes

We proposed absorbing interface conditions for the simulation of linear wave propagation on non-uniform meshes. Based on the superposition principle of second-order linear wave equations, we decompose the interface condition problem into two subproblems around the interface: for the first one the conventional artificial absorbing boundary conditions is applied, while for the second one, the local analytic solutions can be derived. The proposed interface conditions permit a two-way transmission of low-frequency waves across mesh interfaces which can be supported by both coarse and fine meshes, and perform a one-way absorption of high-frequency waves which can only be supported by fine meshes when they travel from fine mesh regions to coarse ones. Numerical examples are presented to illustrate the efficiency of the proposed absorbing interface conditions.     

Special Green’s function boundary element approach for steady-state axisymmetric heat conduction across low and high conducting planar interfaces

The problem of determining the steady-state axisymmetric temperature distribution in a bimaterial with a planar interface is considered here. The interface is either low or high conducting. Special Green’s functions satisfying the thermal conditions on the interface are derived and employed to obtain boundary integral equations whose path of integration does not include the interface. Boundary element procedures that do not require the interface to be discretized into elements are proposed for solving the problem under consideration.