Spectral element solution of steady incompressible viscous free-surface flows

In this paper we present a new approach for the solution of the steady incompressible Navier-Stokes equations in a domain bounded in part by a free surface. In the spatial discretization procedure, a Legendre spectral element method is used to generate the discrete equations. For effective solution of the set of algebraic equations, the geometry is decoupled from the fluid velocity and pressure. In addition, two different algorithms are proposed depending on the importance of surface tension effects. Numerical results are presented to demonstrate the effectiveness of the proposed algorithms.     

Accurate interface-tracking for arbitrary Lagrangian–Eulerian schemes

We present a new method for tracking an interface immersed in a given velocity field which is particularly relevant to the simulation of unsteady free surface problems using the arbitrary Lagrangian–Eulerian (ALE) framework. The new method has been constructed with two goals in mind: (i) to be able to accurately follow the interface; and (ii) to automatically achieve a good distribution of the grid points along the interface. In order to achieve these goals, information from a pure Lagrangian approach is combined with information from an ALE approach. Our implementation relies on the solution of several pure convection problems along the interface in order to obtain the relevant information. The new method offers flexibility in terms of how an “optimal” point distribution should be defined. We have proposed several model problems, each with a prescribed time-dependent velocity field and starting with a prescribed interface; these problems should be useful in order to validate the accuracy of interface-tracking algorithms, e.g., as part of an ALE solver for free surface flows. We have been able to verify first, second, and third order temporal accuracy for the new method by solving these two-dimensional model problems.     

A reduced-basis element methodUne méthode d'éléments en base réduite

Reduced basis methods are particularly attractive to use in order to diminish the number of degrees-of-freedom associated with the approximation to a set of partial differential equations. The main idea is to construct ad hoc basis functions with a large information content. In this Note, we propose to develop and analyze reduced basis methods for simulating hierarchical flow systems, which is of relevance for studying flows in a network of pipes, an example being a set of arteries or veins. We propose to decompose the geometry into generic parts (e.g., pipes and bifurcations), and to construct a reduced basis for these generic parts by considering representative geometric snapshots. The global system is constructed by gluing the individual basis solutions together via Lagrange multipliers. To cite this article: Y. Maday, E.M. Rønquist, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 195–200.     

Approximation of Parametric Derivatives by the Empirical Interpolation Method

We introduce a general a priori convergence result for the approximation of parametric derivatives of parametrized functions. We consider the best approximations to parametric derivatives in a sequence of approximation spaces generated by a general approximation scheme, and we show that these approximations are convergent provided that the best approximation to the function itself is convergent. We also provide estimates for the convergence rates. We present numerical results with spaces generated by a particular approximation scheme—the Empirical Interpolation Method—to confirm the validity of the general theory.     

Regression on parametric manifolds: Estimation of spatial fields,functional outputs,and parameters from noisy data

In this Note we extend the Empirical Interpolation Method (EIM) to a regression context which accommodates noisy (experimental) data on an underlying parametric manifold. The EIM basis functions are computed Offline from the noise-free manifold; the EIM coefficients for any function on the manifold are computed Online from experimental observations through a least-squares formulation. Noise-induced errors in the EIM coefficients and in linear-functional outputs are assessed through standard confidence intervals and without knowledge of the parameter value or the noise level. We also propose an associated procedure for parameter estimation from noisy data.     

Accurate interface-tracking of surfaces in three dimensions for arbitrary Lagrangian–Eulerian schemes

We extend the computational method presented in [1] for tracking an interface immersed in a given velocity field to three spatial dimensions. The proposed method is particularly relevant to the simulation of unsteady free surface problems using the arbitrary Lagrangian–Eulerian framework, and has been constructed with two goals in mind: (i) to be able to accurately follow the interface; and (ii) to automatically maintain a good distribution of the grid points along the interface. The method combines information from a pure Lagrangian approach with information from an ALE approach. The new method offers flexibility in terms of how an “optimal” point distribution should be defined, and relies on the solution of two-dimensional surface convection problems. We verify the new method by solving model problems both in the single and multiple spectral element case, and we compare this method with other traditional alternatives. We have been able to verify first, second, and third order temporal accuracy for the new method by solving these three-dimensional model problems.     

High order numerical approximation of minimal surfaces

We present an algorithm for finding high order numerical approximations of minimal surfaces with a fixed boundary. The algorithm employs parametrization by high order polynomials and a linearization of the weak formulation of the Laplace–Beltrami operator to arrive at an iterative procedure to evolve from a given initial surface to the final minimal surface. For the steady state solution we measure the approximation error in a few cases where the exact solution is known. In the framework of parametric interpolation, the choice of interpolation points (mesh nodes) is directly affecting the approximation error, and we discuss how to best update the mesh on the evolutionary surface such that the parametrization remains smooth. In our test cases we may achieve exponential convergence in the approximation of the minimal surface as the polynomial degree increases, but the rate of convergence greatly differs with different choices of mesh update algorithms. The present work is also of relevance to high order numerical approximation of fluid flow problems involving free surfaces.