Controllability Methods for the Computation of Time-Periodic Solutions; Application to Scattering

Bristeau, Marie-Odile; Glowinski, Roland; Périaux, Jacques

doi:10.1006/jcph.1998.6044

Cited by 52 publications

(94 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The closest numerical method to our own that we have found is due to Bristeau et al (1998), who developed a least squares shooting method for numerical computation of time-periodic solutions of linear dynamical systems with applications in scattering phenomena in two and three dimensions; see also Glowinski and Rossi (2006). These authors employ methods of control theory to compute variational derivatives, and although they only apply their methods to linear problems, they mention that their techniques will also work on non-linear problems.…”

Section: Introductionmentioning

confidence: 99%

Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

Ambrose

Wilkening

2010

J Nonlinear Sci

View full text Add to dashboard Cite

We present a spectrally accurate numerical method for finding nontrivial time-periodic solutions of nonlinear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase, and the real part of one of the Fourier modes at t = 0.We use our method to study global paths of nontrivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our J Nonlinear Sci (2010) 20: 277-308 numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached. By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODEs governing the evolution of solitons using the ansatz suggested by the numerical simulations.

show abstract

Section: Introductionmentioning

confidence: 99%

Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

Ambrose

Wilkening

2010

J Nonlinear Sci

View full text Add to dashboard Cite

show abstract

“…To be more precise, we consider the model, in which the acoustic waves in the fluid domain are modeled by using the velocity potential and the elastic waves in the structure domain are modeled by using displacement. We follow the idea of Bristeau, Glowinski, and Périaux, presented in [3,4,5,6,7], and avoid solving indefinite systems by returning to time-dependent equations. The difference between the initial condition and the terminal condition of the time-dependent system is minimized by an optimization algorithm.…”

Section: Introductionmentioning

confidence: 99%

An optimization-based approach for solving a time-harmonic multiphysical wave problem with higher-order schemes

Mönkölä¹

2013

Journal of Computational Physics

View full text Add to dashboard Cite

An optimization-based approach for solving a time-harmonic multiphysical wave problem with higher-order schemes Mönkölä, Sanna Mönkölä, S. (2013). An optimization-based approach for solving a time-harmonic multiphysical wave problem with higher-order schemes. Journal of Computational Physics , 242 (1 June), 439-459. doi:10.1016/j.jcp.2013.02 AbstractThis study considers developing numerical solution techniques for the computer simulations of time-harmonic fluid-structure interaction between acoustic and elastic waves. The focus is on the efficiency of an iterative solution method based on a controllability approach and spectral elements. We concentrate on the model, in which the acoustic waves in the fluid domain are modeled by using the velocity potential and the elastic waves in the structure domain are modeled by using displacement.Traditionally, the complex-valued time-harmonic equations are used for solving the time-harmonic problems. Instead of that, we focus on finding periodic solutions without solving the time-harmonic problems directly. The time-dependent equations can be simulated with respect to time until a time-harmonic solution is reached, but the approach suffers from poor convergence. To overcome this challenge, we follow the approach first suggested and developed for the acoustic wave equations by Bristeau, Glowinski, and Périaux. Thus, we accelerate the convergence rate by employing a controllability method. The problem is formulated as a least-squares optimization problem, which is solved with the conjugate gradient (CG) algorithm. Computation of the gradient of the functional is done directly for the discretized problem. A graph-based multigrid method is used for preconditioning the CG algorithm.

show abstract

“…Compared to what has been done in Bristeau et al [1998], Glowinski and Lions [1995] the main difficulty is clearly the numerical implementation of the distributed Lagrange multiplier based techniques used to force Dirichlet boundary conditions. We shall consider the forward wave equations only since the backward ones can be treated by similar methods.…”

Section: Solve Next the Following Backward Wave-problemmentioning

confidence: 99%

“…In order to validate the methods discussed in the above sections, we will address the solution of three test problems already solved in Bristeau et al [1998] and Glowinski and Lions [1995] using controllability and obstacle fitted finite element meshes. These problems concern the scattering of planar incident waves by a disk, a convex ogive, and a non-convex reflector (air-intake like).…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…A motivation for using the fictitious domain approach is that it allows-to some extent-the use of uniform meshes, which is clearly an advantage far from the scatters. In order to capture efficiently time-periodic solutions, the fictitious domain methodology is coupled to exact controllability methods close to those utilized in Bristeau et al [1998], Glowinski and Lions [1995]. Various formulations of a wave propagation model problem, including a fictitious domain one, will be discussed in Sections 3 and 4.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Domain Embedding/Controllability Methods for the Conjugate Gradient Solution of Wave Propagation Problems

Chen

Glowinski

Périaux

et al.

Lecture Notes in Computational Science and Engineering

Self Cite

View full text Add to dashboard Cite

Summary. The main goal of this paper is to discuss the numerical simulation of propagation phenomena for time harmonic electromagnetic waves by methods combining controllability and fictitious domain techniques. These methods rely on distributed Lagrangian multipliers, which allow the propagation to be simulated on an obstacle free computational region using regular finite element meshes essentially independent of the geometry of the obstacle and on a controllability formulation which leads to algorithms with good convergence properties to time-periodic solutions. This novel methodology has been validated by the solutions of test cases associated to non trivial geometries, possibly non-convex. The numerical experiments show that the new method performs as well as the method discussed in Bristeau et al. [1998] where obstacle fitted meshes were used.

show abstract

Controllability Methods for the Computation of Time-Periodic Solutions; Application to Scattering

Cited by 52 publications

References 15 publications

Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

An optimization-based approach for solving a time-harmonic multiphysical wave problem with higher-order schemes

Domain Embedding/Controllability Methods for the Conjugate Gradient Solution of Wave Propagation Problems

Contact Info

Product

Resources

About