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Abstract
A time dependent three dimensional acoustic scattering problem is considered. An incoming wave packet is scattered by a bounded simply connected obstacle with locally Lipschitz boundary. The obstacle is assumed to have a constant boundary acoustic impedance. The limit cases of acoustically soft and acoustically hard obstacles are considered. The scattered acoustic field is the solution of an exterior problem for the wave equation. A new numerical method to compute the scattered acoustic field is proposed. This numerical method obtains the time dependent scattered field as a superposition of time harmonic acoustic waves and computes the time harmonic acoustic waves by a new "operator expansion method". That is the time harmonic acoustic waves are solutions of an exterior boundary value problem for the Helmholtz equation. The method used to compute the time harmonic waves improves on the method proposed by Misici, Pacelli, Zirilli [J. Acoust. Soc. Am. 103, 106-113 (1998)] and is based on a "perturbative series" of the type of the one proposed in the operator expansion method by Milder [J. Acoust. Soc. Am. 89, 529-541 (1991)]. Computationally the method is highly parallelizable with respect to time and space variables. Some numerical experiments on test problems obtained with a parallel implementation of the numerical method proposed are shown and discussed from the numerical and the physical point of view.
Abstract
We present a numerical method to compute the solution of a time dependent three dimensional scattering problem for the wave equation. That is given a bounded simply connected obstacle having a known constant acoustic impedance find the scattered wave generated by an incoming wave packet that hits the obstacle. The scattered wave is obtained as the solution of an exterior problem for the wave equation, and is computed as a superposition of time harmonic waves. Each time harmonic wave is computed using the operator expansion method. The method we present is highly parallelizable both with respect to the time and the space variables. In fact the computation of the time harmonic waves can be carried out in parallel and a high level of parallelism can be used in the computation of each time harmonic wave via the operator expansion method. Numerical results on test problems obtained with a parallel implementation of the numerical method proposed here are shown. A discussion of the results both from the numerical and from the physical point of view is given. Some animations of the numerical results obtained are shown.
Abstract
We consider a time dependent three dimensional electromagnetic scattering problem. Let R^{3} be the three dimensional real Euclidean space filled with a medium of electric permittivity e, magnetic permeability m and zero electric conductivity. The quantities e, m are positive constants and there are no free charges in the space. Let W Ì R^{3} be a bounded simply connected obstacle with a locally Lipschitz boundary ¶W, that is assumed to have a nonzero constant boundary electromagnetic impedance. The limit cases of perfectly conducting and perfectly insulating obstacles are studied. We consider an incoming electromagnetic wave packet that hits W. We present a method to compute the scattered electromagnetic field as a superposition of time harmonic electromagnetic waves. These time harmonic electromagnetic waves are the solutions of exterior boundary value problems for the vector Helmholtz equation with the divergence free condition and they are computed with an "operator expansion" method that generalizes the methods presented in [a] and [b]. The "operator expansion" method for time harmonic acoustic scattering from an unbounded surface was introduced for the first time by Milder [c]. The method proposed is computationally very efficient. In fact it is highly parallelizable with respect to time and space variables. Several numerical experiments obtained with a parallel implementation of the method are shown. The numerical results obtained are discussed from the numerical and the physical point of view. The quantitative character of the numerical experience shown is established.
Abstract
In this paper we present a generalization of the "operator expansion method" developed in [a]. Let D contained R^{3} be a bounded simply connected domain with locally Lipschitz boundary. The boundary of D is characterized by a given boundary acoustic impedance not necessarily constant. The operator expansion method has been used to solve the exterior boundary value problem for the Helmholtz equation in R^{3}\D via a "perturbative series". This perturbative series is built using two auxiliary "reference" surfaces. In [a] these surfaces are chosen to be the surfaces of two different spheres, the domain D to be starlike and the series expansion is built using the spherical coordinate system. The main advantage of these choices is that when the basis of the spherical harmonic functions is used to solve the problem the linear system involved in the computation of each term of the perturbative series is a diagonal linear system. However these choices do not guarantee the numerical convergence of the "perturbative series" when the shape of D is "far" from being the shape of a sphere. The new formulation proposed here involves more general reference surfaces, and a more general choice of the coordinate system used to build the expansion than the choice made previously of spherical coordinates. That is domains D, whose boundary is "far" from being the boundary of a sphere, are treated using as reference surfaces two surfaces "close" to the boundary of D. We show the numerical convergence of the associated "perturbative series". The price to be paid for this more general choice of reference surfaces is that a dense linear system must be solved to compute each term of the "perturbative series". To overcome this difficulty a suitable basis of wavelets [b] is used. The use of this basis reduces the solution of the dense linear systems mentioned above to the solution of very sparse linear systems. The numerical method obtained combining these ideas to solve the exterior boundary value problem for the Helmholtz equation is computationally highly parallelizable and is a practical tool to solve the time dependent problem when the wave equation with suitable conditions is considered. In fact the solution of the time dependent acoustic scattering problem can be computed as superposition of the solutions of several time harmonic acoustic scattering problems that is problems for the Helmholtz equation. We show some numerical experiments obtained using a parallel implementation of the computational method proposed. The speed up factor obtained as a function of the number of processors used is shown. Really impressive speed up factors are obtained. The data shown in Fig. 4.4a), that is the "submarine", have been taken from the website: http://avalon.viewpoint.com/. The data that define Obstacle 6 (Fig. 4.4b)) are a slight modification of the data of Fig. 4.4a). Obstacle 6 is one of the obstacles used to test the computational method proposed. We discuss the results obtained both from the quantitative and the qualitative point of view.
Abstract
In this paper we consider a furtivity problem in the context of time dependent three dimensional acoustic obstacle scattering. The scattering problem for a "passive" obstacle is the following: an incoming acoustic wave packet is scattered by a bounded simply connected obstacle with locally Lipschitz boundary having a known boundary acoustic impedance. The scattered wave is the solution of an exterior problem for the wave equation. To make the obstacle furtive we leave "passive" obstacles and we consider "active" obstacles, that is obstacles that when hit by the incoming wave packet react circulating on their boundary a pressure current. The furtivity problem consists in making the acoustic field scattered by the obstacle "as small as possible" choosing a control function, that is a pressure current on the boundary of the obstacle, in the function space of the admissible controls. That is it consists in finding the control function that minimizes a cost functional that will be made precise later. This furtivity problem is of great relevance in many applications.
The mathematical model for this furtivity problem consists in a control problem for the wave equation. That is in the boundary condition for the wave equation on the boundary of the obstacle we introduce a control function that is the so caled pressure current. The cost functional depends on the control function, and on the scattered acoustic field. Note that the scattered field depends on the control function via the boundary conditions. Using the Pontryagin maximum principle we prove that, for a suitable choice of the cost functional, the first order optimality conditions for the furtivity problem considered can be formulated as an exterior problem defined outside the obstacle for a system of two coupled wave equations. To solve this exterior problem we develop a highly parallelizable numerical method based on a "perturbative series" of the type proposed in [a]. This method obtains the time dependent scattered field and the control function as a superposition of time harmonic functions. The space dependent parts of each time harmonic component of the scattered field and of the control function are obtained solving a boundary value problem for two coupled Helmholtz equations. The mathematical model and the numerical method proposed are validated studying numerically some test problems. The results obtained with a parallel implementation of the numerical method proposed on the test problems are shown and discussed from the numerical and the physical point of view. The quantitative character of the results obtained is established.
Abstract
In this paper we study a mathematical model to describe a three dimensional acoustic scattering problem associated to a "vibrating" obstacle that is a bounded simply connected domain contained in the three dimensional real Euclidean space whose shape changes in time. In particular we propose a highly parallelizable numerical method based on the operator expansion method to solve the mathematical model considered. This method makes possible to obtain highly parallelizable algorithms able to compute the solution of the problem considered order by order in perturbation theory, and able to obtain the required solution of the scattering problem summing up the perturbation series. Really impressive speed up factors are observed and reported when the algorithm is executed on the Chiba Cluster, a parallel machine of the Argonne National Laboratory - USA. We validate the mathematical model and the numerical method proposed solving some test problems. The quantitative character of the numerical results obtained is established. The results obtained on the test problems are discussed both from the numerical and the physical point of view. In particular we show that the Doppler spectrum associated to the far field patterns of the scattered acoustic fields depends mainly from the incoming wave and from the excited vibrational modes. To show this we consider the simplest case when the incoming wave is a plane wave and the mechanical vibration is characterized by only one time frequency.
Abstract
We consider ``furtivity" and ``masking" problems in time dependent acoustic obstacle scattering. Roughly speaking a ``furtivity" (``masking") problem consists in making ``undetectable" (``unrecognizable") an object immersed in a medium where an acoustic wave that scatters on the object is propagating. The detection (recognition) of the obstacle must be made through the knowledge of the acoustic field scattered by the object when hit by the propagating wave. These problems are interesting in several application fields. We formulate a mathematical model for the ``furtivity" and ``masking" problems considered consisting in optimal control problems for the wave equation. Using the Pontryagin maximum principle we show that the solution of these control problems can be characterized as the solution of a suitable exterior problem for a system of two coupled wave equations. The numerical solution of these systems involving partial differential equations in four (space, time) independent variables is a critical issue when reliable and efficient procedures to solve the furtivity or masking problem are required. High performance parallel algorithms are desirable to solve these systems. We suggest a computational method well suited for parallel computing and based on the operator expansion method introduced in [a] and developed by the authors and some co-authors in [b]-[f]. Finally we make some comments on the possible extension of this work to electromagnetic obstacle scattering and fluid mechanics.
Abstract
In this paper we consider a masking problem in time dependent three dimensional acoustic obstacle scattering. The masking problem consists in making ``masked" an object W characterized by an acoustic boundary impedance c ³ 0 and immersed in a homogeneous isotropic medium when, hit by an acoustic incident wave, generates a scattered acoustic field. To realize this masking effect we try to make the acoustic field scattered by the obstacle (W;c) "similar" to the field scattered, when hit by the same acoustic incident wave, by a given obstacle D with impedance c^{¢}. We refer to (D;c^{¢}) as the ``mask" and we require that the closure of D is contained in W . We formulate a mathematical model for the masking problem consisting in an optimal control problem for the wave equation. That is we try to make the field scattered by (W;c) similar to the field scattered by the mask (D ;c^{¢}) choosing a control function, that is a pressure current, defined on the boundary of the obstacle for all times, in such a way that a suitable cost functional is minimized. The cost functional depends on the control function, and on the scattered acoustic fields generated by the obstacle (W; c) and by the mask (D;c^{¢}). Using the Pontryagin maximum principle we show that the first order optimality condition for the optimal control problem considered can be formulated as an exterior problem defined outside the obstacle for a system of two coupled wave equations. To solve numerically this exterior problem we develop a highly parallelizable method that is a slight modification of the operator expansion method proposed in [a]. We validate the mathematical model and the numerical method proposed on some test problems and we discuss the results obtained with a parallel implementation of the numerical method from the numerical and the physical point of view.
Abstract
In this paper we consider furtivity and masking problems in time dependent three dimensional electromagnetic obstacle scattering. That is we propose a criterion based on a merit function to ``minimize" or to ``mask" the electromagnetic field scattered by a bounded obstacle when hit by an incoming electromagnetic field and with respect to this criterion we derive the optimal strategy. These problems are natural generalizations to the context of electromagnetic scattering of the furtivity problem in time dependent acoustic obstacle scattering presented in [a]. We propose mathematical models of the furtivity and masking time dependent three dimensional electromagnetic scattering problems that consist in optimal control problems for systems of partial differential equations derived from the Maxwell equations. These control problems are approached using the Pontryagin maximum principle. We formulate the first order optimality conditions for the control problems considered as exterior problems defined outside the obstacle for systems of partial differential equations. Moreover the first order optimality conditions derived are solved numerically with a highly parallelizable numerical method based on a perturbative series of the type considered in [b], [c]. Finally we assess and validate the mathematical models and the numerical method proposed analyzing the numerical results obtained with a parallel implementation of the numerical method in several experiments on test problems. Really impressive speed up factors are obtained executing the algorithms on a parallel machine when the number of processors used in the computation ranges between 1 and 100. Some virtual reality applications and some animations relative to the numerical experiments can be found in this website.
Abstract
Time dependent acoustic scattering problems involving "smart" obstacles are considered. Smart obstacles are obstacles that when hit by an incident acoustic field react in the attempt of pursuing a preassigned goal. Let IR^{3} be the three dimensional real Euclidean space and let W Ì IR^{3} be a bounded simply connected open set with Lipschitz boundary characterized by a constant acoustic boundary impedance c and immersed in an isotropic and homogeneous medium that fills IR^{3}\W. The closure of W will be denoted with `W. When hit by an incident field the obstacle W pursues the preassigned goal circulating on its boundary a suitable pressure current (i.e. a quantity dimensionally given by a pressure divided by a time). The obstacles considered in this paper choose the pressure current to circulate on their boundaries in order to fulfill one of the following goals: i) to be furtive in a given set of the frequency space, ii) to appear in a given set of the frequency space and outside a given set of IR^{3} containing W and W_{G} as similar as possible to a "ghost" obstacle W_{G} having boundary acoustic impedance c_{G}. We assume that `WÇ`W_{G} = Æ and that W_{G} ¹ Æ. The problem corresponding to the first goal will be called definite band furtivity problem, the problem corresponding to the second goal will be called definite band ghost obstacle problem. These goals define two classes of smart obstacles. Let t denote the time variable, the main purpose of this paper is to model the previous problems as optimal control problems for the wave equation introducing a control function (i.e. the pressure current) acting on the boundary of W for t Î IR. The cost functionals proposed depend on the value of the control function on the boundary of the obstacle and on the value of the scattered acoustic field generated by the obstacle on the boundary of the obstacle (in the "furtivity case") or on the boundary of a suitable set containing W and W_{G} (in the "ghost obstacle case"). Under some hypotheses the use of the Pontryagin maximum principle allows us to formulate the first order optimality conditions for the definite band furtivity problem and for the definite band ghost obstacle problem as exterior problems outside the obstacle for a system of two coupled wave equations. Numerical methods to solve these exterior problems are developed. These methods are adapted versions of the method introduced in [a], [b], [c], they belong to the class of the operator expansion methods and are highly parallelizable. Some numerical experiments proving the validity of the control problems proposed as mathematical models of the definite band furtivity problem and of the definite band ghost obstacle problem are shown. The numerical results obtained with a parallel implementation of the numerical methods developed are discussed and their quantitative character is established. The speed up factors obtained using parallel computing are really impressive. This website contains some animations and some virtual reality applications relative to the numerical experiments.
Abstract
We present a highly parallelizable numerical method to solve time dependent acoustic obstacle scattering problems. The method proposed is a generalization of the "operator expansion method" developed in [a]. Using a Fourier transform with respect to time the time dependent scattering problem for the wave equation considered is transformed in an exterior boundary value problem for the Helmholtz equation depending on a parameter. The "operator expansion method" is used to solve the exterior problems for the Helmholtz equation and reduces, via a perturbative approach, the solution of each exterior problem to the solution of a sequence of systems of first kind integral equations defined on a suitable "reference surface". The numerical solution of these systems of integral equations is challenging when scattering problems involving realistic obstacles and wavelengths small compared to the characteristic dimensions of the obstacles are solved. Let R_{T} be the ratio between the characteristic dimension of the obstacle and the wavelength considered. In the numerical experiments presented the greater ratios R_{T} considered range in the interval [13,60]. This implies the use of high dimensional vector spaces to approximate satisfactorily the corresponding systems of integral equations. That is each system of integral equations in the sequence mentioned above after being discretized to be solved numerically becomes a large system of linear equations. Represented on a generic base the matrices that approximate the integral operators after the discretization will be dense matrices. We introduce a new way of using the wavelet transform and new bases of wavelets and we develop a version of the operator expansion method that constructs directly element by element the coefficient matrix of the sparse linear systems that approximate in the wavelet basis considered the systems of integral equations. The resulting numerical method, using affordable computing resources, is able to deal with realistic acoustic scattering problems solving large (sparse) linear systems (up to approximately 5^{.} 10^{5} (real) unknowns and equations in the numerical experience shown here). The computation of the elements of the coefficient matrices of the linear systems considered and several other aspects of the numerical algorithm proposed are highly parallelizable allowing the development of an efficient solver for exterior boundary value problems for the Helmholtz equation. Several numerical experiments involving realistic obstacles and "small" wavelengths are proposed. When possible the quantitative character of the numerical results obtained is established. To evaluate the performance of the proposed algorithm on parallel computing facilities appropriate speed up factors are introduced and evaluated. Some animations and virtual reality applications relative to these numerical experiments can be found in this website.
Abstract
A time harmonic acoustic inverse scattering problem involving smart obstacles is formulated and a method to solve it is proposed. A smart obstacle is an obstacle that when hit by an incoming acoustic wave tries to pursue a given goal circulating a suitable pressure current on its boundary. A pressure current is a quantity whose physical dimension is pressure divided by time. The goals pursued by the smart obstacles that we have considered are the following ones: to be undetectable or to appear with a shape and/or acoustic boundary impedance different from its actual ones eventually in a location in space different from the actual location. The following time harmonic inverse scattering problem is considered: from the knowledge of several far fields generated by the smart obstacle when hit by known time harmonic waves, the knowledge of the goal pursued by the smart obstacle and of its acoustic boundary impedance reconstruct the boundary of the obstacle. A method to solve this inverse problem that generalizes the so called Herglotz function method is proposed. Some numerical experiments that validate the method proposed are presented.
Abstract
In this paper we propose a highly parallelizable numerical method for time dependent acoustic scattering problems involving realistic smart obstacles hit by incoming waves having wavelengths small compared with the characteristic dimension of the obstacles. A smart obstacle is an obstacle that when hit by an incoming wave tries to pursue a goal circulating on its boundary a pressure current. In particular we consider obstacles whose goal is to be undetectable and we refer to them as furtive obstacles. These scattering problems are modelled as optimal control problems for the wave equation. We validate the method proposed to solve the optimal control problem considered on some test problems where a "smart" simplified version of the NASA space shuttle is hit by incoming waves with small wavelengths compared to its characteristic dimension. That is we consider test problems with ratio between the characteristic dimension of the obstacle and wavelength of the time harmonic component of the incoming wave up to approximately one hundred. This website contains animations and virtual reality applications showing some numerical experiments relative to the problems studied in the paper.
Abstract
In this paper we study a time harmonic inverse acoustic scattering problem involving an obstacle that when hit by an incoming acoustic wave tries to appear in a location in space different from its actual location eventually with a shape and an acoustic boundary impedance different from its actual ones. We refer to this problem as inverse ghost obstacle problem and to the scatterers of this type as a special class of smart obstacles. The term "ghost" comes from the fact that the smart obstacle, when hit by the incoming wave, tries to appear as a ghost obstacle, that is as a virtual obstacle in a location in space where no real obstacle is present. We assume that the intersection between the obstacle and the ghost obstacle is empty. The obstacle tries to produce this virtual image of itself circulating on its boundary a suitable pressure current. A pressure current is a quantity whose physical dimension is pressure divided by time. The following time harmonic inverse ghost obstacle scattering problem is considered: from the knowledge of several far fields generated by the smart obstacle when hit by known time harmonic waves, of its acoustic boundary impedance and of the acoustic boundary impedance of the ghost obstacle find the location of the smart obstacle, that is find a point in the interior of the smart obstacle and eventually find the shape of the smart obstacle. The method proposed to solve this problem is a generalization of the well known Herglotz function method. We present some numerical experiments based on synthetic data that are used to test the numerical method introduced here.