Iterative solution of the Lippmann-Schwinger equation in strongly scattering acoustic media by randomized construction of preconditioners

Summary In this work the Lippmann-Schwinger equation is used to model seismic waves in strongly scattering acoustic media. We consider the Helmholtz equation, which is the scalar wave equation in the frequency domain with constant density and variable velocity, and transform it to an integral equation of the Lippmann-Schwinger type. To directly solve the discretized problem with matrix inversion is time-consuming, therefore we use iterative methods. The Born series is a well-known scattering series which gives the solution with relatively small cost, but it has limited use as it only converges for small scattering potentials. There exist other scattering series with preconditioners that have been shown to converge for any contrast, but the methods might require many iterations for models with high contrast. Here we develop new preconditioners based on randomized matrix approximations and hierarchical matrices which can make the scattering series converge for any contrast with a low number of iterations. We describe two different preconditioners; one is best for lower frequencies and the other for higher frequencies. We use the fast Fourier transform (FFT) both in the construction of the preconditioners and in the iterative solution, and this makes the methods efficient. The performance of the methods are illustrated by numerical experiments on two 2D models.

A note on the fast direct method for discrete elliptic problems

We consider a method due to P. Vassilevski and Yu. A. Kuznetsov [4, 10] for solving linear systems with matrices of low Kronecker rank such that all factors in Kronecker products are banded. Most important examples of such matrices arise from discretized div K grad operator with diffusion term k1(x)k2(y)k3(z). Several practical issues are addressed: an MPI implementation with distribution of data along processor grid inheriting Cartesian 3D structure of discretized problem; implicit deflation of the known nullspace of the system matrix; links with two-grid framework of multigrid algorithm which allow one to remove the requirement of Kronecker structure in one or two of axes. Numerical experiments show the efficiency of 3D data distribution having the scalability analogous to (structured) HYPRE solvers yet the absolute timings being an order of magnitude lower, on the range from 10 to 104 cores.

SIMULATION OF A SPILLED OIL SLICK WITH A SHALLOW WATER MODEL WITH FREE BOUNDARY

In this paper we present a new approach to describe the behaviour of a pollutant slick at the sea surface. To this end, we consider that the pollutant and the water are immiscible and we propose a two-layer model where the lower layer corresponds to the water and the upper layer represents the pollutant. Since the dimension of the pollutant slick is generally much smaller than the domain occupied by the sea, we propose to compute the motion of the pollutant with a shallow water model with free boundary only in the domain occupied by the pollutant. To discretize in time the problem with free boundary, we use an ALE formulation coupled with the characteristic method. Then, to solve the space discretized problem, we approximate the pollutant velocity by using a Galerkin method with a special basis which verifies the boundary conditions and simplifies significantly the resolution. Finally we test this work in a real situation: the dam of Calacuccia (Corsica).

THEORETICAL AND NUMERICAL STUDY OF AN IMPLICIT DISCRETIZATION OF A 1D INVISCID MODEL FOR RIVER FLOWS

In this paper, we consider a 1D inviscid model to simulate river flows whose unknown are the water elevation, the flow rate and the wet area. We rewrite this model in terms of the wet area and the flow rate. Next, by using a weak formulation and a discretization by characteristics, we derive an implicit semidiscretized scheme that involves a variational inequality. Under suitable hypothesis, we prove that this scheme has at least one solution. We present an algorithm for the numerical solution of the fully discretized problem and the numerical results obtained for channels with parabolic and rectangular sections and variable depth. Finally, we perform a stability analysis for the semidiscretized scheme applied to a similar problem where the convection term has been omitted.

Stepwise Loading of Half-Spaces in Elliptical Contact

A new analytical solution for Hertzian surfaces in contact under stepwise oblique loading is presented in this paper. It is written as a superposition of so-called Catta-neo-Mindlin functions, which represent the tangential stress distribution for constant normal and monotonically increasing tangential forces. The size of the stick zone is determined by cones in the force space, which are also known as yield cones. It has a simpler form than the differential force-displacement relations of plasticity theory and the numerical discretization of the influence integrals. A short computer algorithm is proposed and compared with a numerical solution of the discretized problem.

Three-Dimensional Unilateral Frictionless Contact Problem for Finite Bodies

The numerical solution of the three-dimensional frictionless contact problem is obtained by means of a boundary element discretization of a variational inequality and its related extremum principle. The discretization leads to a finite dimensional quadratic programming problem, solved by a modification of the gradient projection method. The associated Green’s function is approximated using a standard direct boundary element procedure. The numerical method is applicable to any type of contacting bodies geometry under arbitrary loading. The examples considered were chosen such as to illustrate a distinct ability of the method to capture the influence of a body shape on the contact area and pressure acting in it. It has been demonstrated that the symmetry properties of the Green’s operator hold only asymptotically for the discretized problem.

Discretizations for the average impulse control of piecewise deterministic processes

This paper presents a state space and time discretization for the general average impulse control of piecewise deterministic Markov processes (PDPs). By combining several previous results we show that under some continuity, boundedness and compactness conditions on the parameters of the process, boundedness of the discretizations, and compactness of the state space, the discretized problem will converge uniformly to the original one. An application to optimal capacity expansion under uncertainty is given.

Discretizations for the average impulse control of piecewise deterministic processes

This paper presents a state space and time discretization for the general average impulse control of piecewise deterministic Markov processes (PDPs). By combining several previous results we show that under some continuity, boundedness and compactness conditions on the parameters of the process, boundedness of the discretizations, and compactness of the state space, the discretized problem will converge uniformly to the original one. An application to optimal capacity expansion under uncertainty is given.

Optimum Circulation Distributions for a Class of Marine Propulsors

A new method for determining the optimum circulation distributions for both single- and multiple-stage marine propulsors is developed. The lifting-line model for the propulsor is first discretized with a vortexlattice. Variational calculus is then applied to this discretized problem. The result is a general procedure for determining optimum circulation distributions. This procedure can be readily extended to increasingly complex combinations of interacting lifting lines. The propeller lifting-line, vortex-lattice model is described in some detail. Circulation optimization equations for a propeller are derived. The equations are shown to recover traditional results for both light and moderate propeller loading, as well as for wake adapted propellers. Optimization equations for the torque limited case are presented. The vortex-lattice model for multiple stage propulsors is described and the optimization equations are derived. Examples of optimum circulation distributions for contrarotating propellers, a vane-wheel propulsor, and a propeller with pre-swirl stator are presented.