Controllability method for acoustic scattering with spectral elements

Self 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.

Section: Time Discretizationmentioning

confidence: 99%

Section: Objective Functionalmentioning

confidence: 99%

See 1 more Smart Citation

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

Mönkölä¹

2013

Self Cite

“…For this purpose we use a controllability algorithm [19,20,21,22]. The main idea of the algorithm is to return to the time dependent wave equation and find initial conditions such that after one time-period the solution and its time derivative coincide with the initial conditions.…”

Section: Introductionmentioning

confidence: 99%

Time-harmonic elasticity with controllability and higher-order discretization methods

Mönkölä

Heikkola

Pennanen

et al. 2008

Self Cite

The time-harmonic solution of the linear elastic wave equation is needed for a variety of applications. The typical procedure for solving the time-harmonic elastic wave equation leads to difficulties solving large-scale indefinite linear systems. To avoid these difficulties, we consider the original time dependent equation with a method based on an exact controllability formulation. The main idea of this approach is to find initial conditions such that after one time-period, the solution and its time derivative coincide with the initial conditions. The wave equation is discretized in the space domain with spectral elements. The degrees of freedom associated with the basis functions are situated at the Gauss-Lobatto quadrature points of the elements, and the Gauss-Lobatto quadrature rule is used so that the mass matrix becomes diagonal. This method is combined with the second-order central finite difference or the fourth-order Runge-Kutta time discretization. As a consequence of these choices, only matrix-vector products are needed in time dependent simulation. This makes the controllability method computationally efficient.

“…In [38], we used the central finite difference scheme for time discretization. That scheme is second order accurate and with a diagonal mass matrix also fully explicit, which are both essential properties for computational efficiency.…”

Section: Introductionmentioning

confidence: 99%

Controllability method for the Helmholtz equation with higher-order discretizations

Heikkola¹,

Mönkölä²,

Pennanen³

et al. 2007

Self Cite

We consider a controllability technique for the numerical solution of the Helmholtz equation. The original time-harmonic equation is represented as an exact controllability problem for the time-dependent wave equation. This problem is then formulated as a least-squares optimization problem, which is solved by the conjugate gradient method. Such an approach was first suggested and developed in the 1990s by French researchers and we introduce some improvements to its practical realization.We use higher-order spectral elements for spatial discretization, which leads to high accuracy and lumped mass matrices. Higher-order approximation reduces the pollution effect associated with finite element approximation of time-harmonic wave equations, and mass lumping makes explicit time-stepping schemes for the wave equation very efficent. We also derive a new way to compute the gradient of the least-squares functional and use algebraic multigrid method for preconditioning the conjugate gradient algorithm.Numerical results demonstrate the significant improvements in efficiency due to the higher-order spectral elements. For a given accuracy, spectral element method requires fewer computational operations than conventional finite element method. In addition, by using higher-order polynomial basis the influence of the pollution effect is reduced.