V.S. Ryaben'kii

Active Shielding and Control of Noise

We present a mathematical framework for the active control of time-harmonic acoustic disturbances. Unlike many existing methodologies, our approach provides for the exact volumetric cancellation of unwanted noise in a given predetermined region of space while leaving unaltered those components of the total acoustic field that are deemed friendly. Our key finding is that for eliminating the unwanted component of the acoustic field in a given area, one needs to know relatively little; in particular, neither the locations nor structure nor strength of the exterior noise sources need to be known. Likewise, there is no need to know the volumetric properties of the supporting medium across which the acoustic signals propagate, except, perhaps, in the narrow area of space near the boundary (perimeter) of the domain to be shielded. The controls are built based solely on the measurements performed on the perimeter of the region to be shielded; moreover, the controls themselves (i.e., additional sources) are also c...

Artificial boundary conditions for the numerical solution of external viscous flow problems

In this paper we describe an algorithm for the nonlocal artificial boundary conditions setting at the external boundary of a computational domain while numerically solving unbounded viscous compressible flow problems past the finite bodies. Our technique is based on the usage of generalized Calderon projection operators and the application of the difference potentials method. Some computational results are presented.

Global discrete artificial boundary conditions for time-dependent wave propagation

We construct global artificial boundary conditions (ABCs) for the numerical simulation of wave processes on unbounded domains using a special non-deteriorating algorithm that has been developed previously for the long-term computation of wave-radiation solutions. The ABCs are obtained directly for the discrete formulation of the problem; in so doing, neither a rational approximation of ``non-reflecting kernels, nor discretization of the continuous boundary conditions is required. The extent of temporal nonlocality of the new ABCs appears fixed and limited; in addition, the ABCs can handle artificial boundaries of irregular shape on regular grids with no fitting/adaptation needed and no accuracy loss induced. The non-deteriorating algorithm, which is the core of the new ABCs, is inherently three-dimensional, it guarantees temporally uniform grid convergence of the solution driven by a continuously operating source on arbitrarily long time intervals, and provides unimprovable linear computational complexity with respect to the grid dimension. The algorithm is based on the presence of lacunae, i.e., aft fronts of the waves, in wave-type solutions in odd-dimension spaces. It can, in fact, be built as a modification on top of any consistent and stable finite-difference scheme, making its grid convergence uniform in time and at the same time keeping the rate of convergence the same as that of the non-modified scheme. In the paper, we delineate the construction of the global lacunae-based ABCs in the framework of a discretized wave equation. The ABCs are obtained for the most general formulation of the problem that involves radiation of waves by moving sources (e.g., radiation of acoustic waves by a maneuvering aircraft). We also present systematic numerical results that corroborate the theoretical design properties of the ABCs algorithm.

Inverse source problem and active shielding for composite domains

The problem of active shielding (AS) for a multiply connected domain consists of constructing additional sources of the field (e.g., acoustic) so that all individual subdomains can either communicate freely with one another or otherwise be shielded from their peers. This problem can be interpreted as a special inverse source problem for the differential equation (or system) that governs the field. In the paper, we obtain general solution for a discretized composite AS problem and show that it reduces to solving a collection of auxiliary problems for simply connected domains. c

Experimental Validation of the Active Noise Control Methodology Based on Difference Potentials

To achieve active noise cancellation over a large area, it is often necessary to get a measure of the physical properties of the noise source to devise a counter measure. This, however, is not practical in many cases. A mathematical approach, the Difference Potential Method, can provide an alternative solution for active shielding over a large area. In this approach, the cancellation of unwanted noise requires only measurements near the boundary surface but not at the source itself, and it does not require any other information. Moreover, the solution based on difference potentials applies to bounded domains in the presence of acoustic sources inside the domain to be shielded. This paper reports on the results of experimental validation. It has been demonstrated that while preserving the wanted sound, the developed approach can cancel out the unwanted noise. The volumetric noise cancellation offered by the proposed methodology along with leaving the wanted sound unchanged is a unique feature compared to other techniques available in the literature. It can be most useful in the context of applications related to civil aviation, in particular, for eliminating the exterior noise inside the passenger compartments of both current and future generation of commercial aircraft.

Active control of sound with variable degree of cancellation

Abstract We formulate and solve a control problem for the field (e.g.,xa0time-harmonic sound) governed by a linear PDE or system on a composite domain in R n . Namely, we require that simultaneously and independently on each subdomain the sound generated in its complement be attenuated to a desired degree. This goal is achieved by adding special control sources defined only at the interfacexa0between the subdomains. We present a general solution for controls in the continuous and discrete setting.

An effective numerical technique for solving a special class of ordinary difference equations

Abstract We consider a system of ordinary difference equations with constant coefficients, which is defined on an infinite one-dimensional mesh. The right-hand side (RHS) of the system is compactly supported, therefore, the system appears to be homogeneous outside some finite mesh interval. At infinity, we impose certain boundary conditions, e.g., conditions of boundedness or decay of the solution, so that the resulting boundary-value problem is uniquely solvable and well posed. We also consider a truncation of this infinite-domain problem to some finite mesh interval that entirely contains the support of the RHS. We require that the solution to this truncated problem, which is the one we are going to actually calculate, coincides on the finite mesh interval where it is defined with the corresponding fragment of the solution to the original (infinite) problem. This requirement necessitates setting some special boundary conditions at the ends of the aforementioned finite interval. In so doing, one should guarantee an exact transfer of boundary conditions from infinity through the (semi-infinite) intervals of homogeneity of the original system. It turns out that the desired boundary conditions at the ends of the finite interval can be naturally formulated in terms of the eigen subspaces of the system operator. This, in turn, enables us to develop an effective numerical algorithm for solving the system of ordinary difference equations on the finite mesh interval. This algorithm can be referred to as a version of the well-known successive substitution technique but without its final (“inverse” or “resolving”) stage. The special class of systems described in this paper appears to be most useful when constructing highly accurate artificial boundary conditions (ABCs) for the numerical treatment of problems initially formulated on unbounded domains. Therefore, an effective numerical algorithm for solving such systems becomes an important issue.