Finite Element Methods in Local Active Control of Sound

Bermúdez, Alfredo; Gamallo, Pablo; Rodrı́guez, Rodolfo

doi:10.1137/s0363012903431785

Cited by 36 publications

(59 citation statements)

References 14 publications

(19 reference statements)

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“…If in addition, finitely many pointwise state constraints are imposed in points inside the domain, that do not coincide with the points where the Dirac delta measures are concentrated, then the problem can be also written as a nonlinear programming problem in finite dimensions and the results presented in our paper can also be applied. The error estimate for the finite element approximation obtained in [4] for the linear state equation is the same as ours. Therefore, under natural assumptions we would arrive at the error estimate of order h 2 | log(h)| for the optimal point controls.…”

Section: Introductionmentioning

confidence: 89%

“…Then, no interpolation error of the optimal control occurs that in the case of control functions limits the order of convergence to h for a cellwise constant approximation of the control variable, see, e.g., [3], and to h 3/2 for a piecewise linear approximation, see, e.g., [9,30]. In [4], an optimal control problem of sound is considered, where the controls are chosen to be finite Dirac delta measures. If in addition, finitely many pointwise state constraints are imposed in points inside the domain, that do not coincide with the points where the Dirac delta measures are concentrated, then the problem can be also written as a nonlinear programming problem in finite dimensions and the results presented in our paper can also be applied.…”

Section: Introductionmentioning

confidence: 99%

“…The authors are grateful to W. Alt (Jena) and D. Klatte (Zurich) for their helpful comments and suggestions. Moreover, we thank the referee who draw our attention to the paper [4].…”

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confidence: 99%

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Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space

Merino

Tröltzsch²,

Vexler³

2009

ESAIM: M2AN

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Abstract.The finite element approximation of optimal control problems for semilinear elliptic partial differential equation is considered, where the control belongs to a finite-dimensional set and state constraints are given in finitely many points of the domain. Under the standard linear independency condition on the active gradients and a strong second-order sufficient optimality condition, optimal error estimates are derived for locally optimal controls.Mathematics Subject Classification. 49J20, 35B37.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

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Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space

Merino

Tröltzsch²,

Vexler³

2009

ESAIM: M2AN

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“…The mathematical side of these problems have been considered in [7,11,12], for example. Approaches using nite element modeling, are presented in articles [19,5,17]. In [19], resonance modes for mining vehicle are studied by modal coupling analysis and antinoise is optimized by using FEM model to obtain global noise control in the cabin.…”

Section: Introductionmentioning

confidence: 99%

“…In [19], resonance modes for mining vehicle are studied by modal coupling analysis and antinoise is optimized by using FEM model to obtain global noise control in the cabin. In [5], a local active noise control method based on the nite element method is described which minimizes noise locally in microphone locations. A method to determine the optimal locations for antinoise actuators is also presented.…”

Section: Introductionmentioning

confidence: 99%

Local Control of Sound in Stochastic Domains Based on Finite Element Models

2011

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A numerical method for optimizing the local control of sound in a stochastic domain is developed. A three-dimensional enclosed acoustic space, for example, a cabin with acoustic actuators in given locations is modeled using the nite element method in the frequency domain. The optimal local noise control signals minimizing the least square of the pressure eld in the silent region are given by the solution of a quadratic optimization problem. The developed method computes a robust local noise control in the presence of randomly varying parameters such as variations in the acoustic space. Numerical examples consider the noise experienced by a vehicle driver with a varying posture. In a model problem, a signicant noise reduction is demonstrated at lower frequencies.

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Numerical analysis for the approximation of optimal control problems with pointwise observations

Chang

Gong

Yan

2013

Math Methods in App Sciences

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Communicated by Q. WangIn this paper, we study the numerical methods for optimal control problems governed by elliptic PDEs with pointwise observations of the state. The first order optimality conditions as well as regularities of the solutions are derived. The optimal control and adjoint state have low regularities due to the pointwise observations. For the finite dimensional approximation, we use the standard conforming piecewise linear finite elements to approximate the state and adjoint state variables, whereas variational discretization is applied to the discretization of the control. A priori and a posteriori error estimates for the optimal control, the state and adjoint state are obtained. Numerical experiments are also provided to confirm our theoretical results.of elliptic equations with measure data presented in [16] and [17], a priori error estimates for the control, the state, and the adjoint state are derived. Moreover, the a posteriori error estimates of residual type are presented, which can be applied as indicators for the mesh refinement in adaptive finite element methods. Numerical examples are also included to illustrate the theoretical results. It should be pointed out that our theoretical and numerical results are still valid when the penalty parameter˛is very small, which often appears in many applications, for example, inverse problems.Our paper is organized as follows. In the next section, we present the mathematical setting and formulate the optimal control problems in detail. Finite element approximation to optimal control problems is also presented. In Sections 3 and 4, we prove the a priori error and a posteriori error estimates, respectively. We finally present numerical results, which confirm our analytical findings in section 5.where B is the linear operator from U to L 2 . /, U is the control space. Here, B can be an identity operator such that Bu D u or Bu D ! u where ! is an open subset of and ! the characteristic function of the subset !. z D .z 1 , , z m / 2 R m is given, and X j (j D 1, , m) are space points strictly contained in . Moreover, U ad U is a closed convex subset of typeAccording to the assumptions on and A, for each Bu 2 L 2 . /, we have y.x; u/ 2 H 2 . / \ H 1 0 . /, thus y.x; u/ 2 C. / by imbedding theorem and the aforementioned optimal control problem is well-defined. From standard arguments ([18]), we can prove that the

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Finite Element Methods in Local Active Control of Sound

Cited by 36 publications

References 14 publications

Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space

Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space

Local Control of Sound in Stochastic Domains Based on Finite Element Models

Numerical analysis for the approximation of optimal control problems with pointwise observations

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