Models of elastic plates with piezoelectric inclusions part I: Models without homogenization

Canon, E.; Lenczner, Michel

doi:10.1016/s0895-7177(97)00159-3

Cited by 13 publications

(26 citation statements)

References 14 publications

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“…We are going to use a simplified model for the cathode, one can refer to [18] for a more sophisticated model (that will be the subject of future work) or [2,10] for equivalent approaches . The electric charge on the ith cathode reads We consider that each pair of electrodes (one cathode and one anode) are connected to a generator with an internal resistance (see Fig.…”

Section: Source Modelingmentioning

confidence: 99%

Mathematical and numerical modelling of piezoelectric sensors

Imperiale

Joly

2012

ESAIM: M2AN

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Abstract. The present work aims at proposing a rigorous analysis of the mathematical and numerical modelling of ultrasonic piezoelectric sensors. This includes the well-posedness of the final model, the rigorous justification of the underlying approximation and the design and analysis of numerical methods. More precisely, we first justify mathematically the classical quasi-static approximation that reduces the electric unknowns to a scalar electric potential. We next justify the reduction of the computation of this electric potential to the piezoelectric domains only. Particular attention is devoted to the different boundary conditions used to model the emission and reception regimes of the sensor. Finally, an energy preserving finite element/finite difference numerical scheme is developed; its stability is analyzed and numerical results are presented.Mathematics Subject Classification. 35L05, 35A35, 73R05, 35A40.

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Section: Source Modelingmentioning

confidence: 99%

Mathematical and numerical modelling of piezoelectric sensors

Imperiale

Joly

2012

ESAIM: M2AN

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“…Now, let us derive the inequality (11). Let us denote by Θ 1 2 and Θ 1 3 (respectively Θ 2 2 and Θ 2…”

Section: 1mentioning

confidence: 99%

“…On the one hand, we have already derived models for elastic plates and shells including a great number of periodically distributed piezoelectric transducers and distributed electronics in specific configurations; see Canon and Lenczner 10,11 and Senouci-Bereksi and Lenczner. 23 On the other hand, a general model for periodically distributed electronic network including resistors, current sources and voltage sources was announced in Ref.…”

Section: Introductionmentioning

confidence: 99%

Homogenization of Electrical Networks Including Voltage-to-Voltage Amplifiers

Lenczner

Senouci-Bereksi

1999

Math. Models Methods Appl. Sci.

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We derive the homogenized model of periodic electrical networks which includes resistive devices, voltage-to-voltage amplifiers, sources of tension and sources of current. On the one hand, in considering the homogenized problem, general conditions are stated insuring the existence and uniqueness of the solution. They are formulated in function of the network topology. On the other hand, the two-scale transformation introduced by Arbogast, Douglas and Hornung is adapted to the context of electrical networks. New two-scale convergence results, inspired by the principle of Allaire's two-scale convergence, are shown in this context. In particular, the two-scale convergence for the tangential derivative on a network is established. Following these results, two models of homogenized networks are derived. The first one belongs to a general framework whereas the second one does not. 902 M. Lenczner & G. Senouci-Bereksi the period. We make some assumptions about solution estimates. This choice is led by its interest in applications and by its relative simplicity. In particular, we assume that the amplifier's coefficients are of zero order with respect to ε. The general model related to this framework is stated in Theorem 3. Finally, a particular example with coefficients at the order ε −1 is treated in Theorem 4.Let us note that a homogenized model of two-dimensional electrical networks made of resistors have already been derived in Ref. 25. The method developed by Vogelius was based on an extension of the solution to an open set, which includes the electrical network. The proofs were based on some finite element techniques. The technical difference between our approach and that of Ref. 25 is that, no extension of the solution is required, and the proofs are valid for a network imbedded in an n-dimensional Euclidean space where n ≥ 1. In addition, voltage sources, current sources and voltage-to-voltage amplifiers are taken into account in our approach. This was not the case in Ref. 25. . 15 The approach of Refs. 12 and 13 is based on an asymptotic analysis where both the beam thickness and the truss period lengths vanish. D. Caillerie and Al. introduced the discrete homogenization method for the same problem. In this approach, the unknown are displacement of vertices and tensions of the edges. The model derivation is based on an asymptotic expansion of the solution.The paper is divided into eight sections. In Sec. 2, we will consider an electrical network including resistors, tension sources, current sources and voltage-to-voltage amplifiers. We will provide a set of conditions on the network topology under which the problem is well-posed. In Sec. 3, two-scale convergence results concerning functions defined on electrical networks will be explained. In Sec. 4, a general framework for the homogenization of electric network based on the results of Secs. 2 and 3 will be detailed. Then, a particular example of electric network not belonging to the general framework will be described, and its homogenized model stated. In Sec...

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“…This method was also successfully applied in [5,8,9,13,36,48] for deriving two-dimensional models for thin piezoelectric plates. Extensions of this method for thin piezoelectric shells can be found in [25,33,34].…”

Section: Introductionmentioning

confidence: 99%

A model for bending and stretching of piezoelectric rods obtained by asymptotic analysis

Viaño

Figueiredo²,

Ribeiro³

et al. 2014

Z. Angew. Math. Phys.

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The aim of this paper was to obtain a new model for the bending-stretching of an anisotropic heterogeneous linearly piezoelectric cantilever rod when the electric potential is applied on the both ends. The process is assumed to be static, and the piezoelectric material is monoclinic of class 2. To derive the model, we start with the corresponding threedimensional problem, introduce a change of variable together with a scaling of the unknowns and then we use a passage to the limit procedure, based on arguments of asymptotic analysis taking the diameter of the cross-section as small parameter. Finally, we prove a result of strong convergence that justifies both the method and the one-dimensional model obtained. One of the most relevant features of this one-dimensional model is that the stretching is coupled with the electric potential, while the bendings are not.Mathematics Subject Classification (2010). Primary 74K10; Secondary 41A60 · 35C20 · 35Q60.

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Models of elastic plates with piezoelectric inclusions part I: Models without homogenization

Cited by 13 publications

References 14 publications

Mathematical and numerical modelling of piezoelectric sensors

Mathematical and numerical modelling of piezoelectric sensors

Homogenization of Electrical Networks Including Voltage-to-Voltage Amplifiers

A model for bending and stretching of piezoelectric rods obtained by asymptotic analysis

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