A piezoelectric solid shell element based on a mixed variational formulation for geometrically linear and nonlinear applications

TL;DR: In this paper, a variational principle of the Hu-Washizu type is used to model a piezoelectric solid shell finite element formulation, which allows large deformations and includes stability problems.

About: This article is published in Computers & Structures.The article was published on 2008-01-01 and is currently open access. It has received 69 citations till now. The article focuses on the topics: Finite element method & Electric potential.

Summary

1. INTRODUCTION

• The main control requirement in a solar power plant is to maintain the outlet oil temperature of the collector field at a constant pre-specified value.
• Therefore, it is difficult to obtain a satisfactory performance over the whole operating range with a fixed linear controller.
• By means of a Lyapunov analysis a stability condition for the weights updating is employed.

2. THE SOLAR POWER PLANT

• The Acurex distributed solar collector field at Plataforma Solar de Almería (PSA) is quite well described in available literature (Kaltz, 1982; Camacho et al, 1992) and is located at the desert of Tabernas, in south of Spain.
• The field consists of 480 distributed solar collectors arranged in 20 rows, which form 10 parallel loops.
• The cold inlet oil is collected from the bottom of the storage tank and is passed through the field by using a pump at the field inlet.
• The heated fluid is introduced into the storage tank to be used for electrical energy generation or feeding a heat exchanger of the desalination plant.

3. RECURRENT NEURAL NETWORKS

• 1 Proposed Recurrent Neural Architecture Given the approximation capabilities of RNN (Jin et al, 1999) it is assumed that there exist a RNN, described by (2) and shown in Fig. 2, that is able to describe the plant’s dynamics.
• On the other hand, this can be seen as a hybrid model, with a linear and a non-linear part.
• Several training algorithms have been proposed to recursively adjust the network parameters in recurrent networks, also known as On-line learning.
• To this aim it is assumed that the matrices A and C are static (off-line evaluated) and only the matrices B and D are to be updated on-line.

4. NON-LINEAR OUTPUT REGULATION

• The OR problem for linear systems was solved by Francis (1977).
• For non-linear discrete time systems, Castillo et al (1993), using the zero output constrained algorithm (Monaco and Normand-Cyrot, 1987), has shown that the solution for the problem is reduced to the solution of transcendental non-linear equations, which represent the discrete time counterpart of the differential and transcendental equations, found for the continuous time systems by Isidori and Byrnes (1990).
• As given by equation (12), the solution of the output regulator problem is reduced to a set of non-linear difference equations, known as regulator equations.
• Based on a Taylor series expansion as well, (Huang and Rugh, 1992) have proposed an approximation method for the continuous case.
• The block diagram of the proposed control structure is shown in Fig.

5. EXPERIMENTAL RESULTS

• The experiments were carried out in the Acurex Solar Collectors Field of the Plataforma Solar de Almería on 14 and 15 June 2001.
• The effectiveness of the developed approach was first tested using a non-linear distributed parameter model of the Acurex field, developed at the University of Sevilla (Berenguel et al, 1993).
• Fig. 5 shows the off-line modelling results considering the experimental data set collected on 14 June 2001.
• In Fig. 5 the good performance of the learning methodology is clearly illustrated.
• The disturbance rejection capabilities of the controller are also acceptable, shown by a change in the inlet oil temperature, carried out at instant 15h00m.

6. CONCLUSIONS

• A non-linear control scheme based on a recurrent neural network has been implemented in real-time and applied to a distributed collector field in a solar power plant.
• The process is characterised by different operating conditions, depending on the changes in dynamics caused by variations in the solar radiation, reference temperature and plant characteristics.
• The proposed strategy is a systematic one, which can be easily applied to a wide variety of processes with a small initial knowledge of the plant model.
• The neural model can adaptively learn the system uncertainties and the regulator law adjusts the control action in order to guarantee a robust asymptotic error convergence.
• The simplicity and reliability of neuro-control gives high potential for the development of efficient and intelligent control systems.

Figures

Table 2: Solutions for displacements and electric potential in membrane patch test

Table 3: Solutions for displacements and electric potential in bending patch test

Figure 5: Telescopic actuator: system, boundary conditions and finite element model

Table 1: Simplified material properties; not listed parameters are zero

Figure 1: Element mesh for the patch tests and interior nodes 1, 2, 3, 4 on top of the surface

Table 6: Telescopic actuator: material properties

Table 4: Material properties of the plate

Figure 4: Load deflection curve and plots of the vertical deflection at characteristic points

Table 5: Critical electric potential ϕ [V] for the first four buckling modes

Figure 7: Electric potential versus axial deflections of the different cylinders

Figure 3: Applied electric potential and boundary conditions

Figure 2: Finite element model of the laminated square plate with the stacking sequence of Graphite Epoxy [0o, 90o, 90o, 0o]

Figure 6: Telescopic actuator: deformed configuration with a plot of the electric potential and the axial displacement

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "A piezoelectric solid shell element based on a mixed variational formulation for geometrically linear and nonlinear applications" ?

The paper is focused on a piezoelectric solid shell finite element formulation. The formulation is based on a variational principle of the Hu-Washizu type including six independent fields: displacements, electric potential, strains, electric field, mechanical stresses and dielectric displacements. The element has 8 nodes with displacements and the electric potential as nodal degrees of freedom. 

Q2. What is the simplest way to obtain correct results in bending dominated situations?

A bilinear distribution through the thickness of the electric field is assumed to obtain correct results in bending dominated situations. 

Q3. What is the main principle of the Hu-Washizu type?

The formulation is based on a variational principle of the Hu-Washizu type including six independent fields: displacements, electric potential, strains, electric field, mechanical stresses and dielectric displacements. 

Q4. What is the common assumption in piezoelectric models?

A common assumption in piezoelectric models is that the electric field is constant through the thickness inside the actuator or sensor. 

Q5. What is the general variational formulation principle?

The finite element formulation is based on the most general variational formulation principle of the Hu-Wahizu type and includes six independently assumed field variables: the stress field, the strain field, the displacements, the electric displacements, the electric field, and the electric potential. 

Q6. How many elements are applied to each cylinder?

For each cylinder 5 elements in axial direction, 1 element through the thickness and 12 elements in circumferential directions are applied. 

Q7. What is the main reason why the solid shells have laminate forms?

Due to the fact that the piezoelectric devices have traditionally laminate forms, the above mentioned shell formulations include a more or less sophisticated laminate theory. 

Q8. Why are the material constants in thickness not required in the plate formulation of [16]?

Due to the fact that the material constants in thickness direction are not required in the plate formulation of [16], they are assumed in the present work. 

Q9. What is the simplest method of constructing a solid shell?

The solid shell elements circumvent complicated laminate theories by modeling each ply in a laminate with one element in thickness direction. 

Q10. What is the metric coefficient of the electric field?

The covariant components of the electric field are also arranged in a vector Ec = [ E1, E2, E3] T withEi = − ∂ϕ ∂ξi , (3)here ϕ denotes the electric potential. 

Q11. What is the buckling behavior of a graphite epoxy plies?

2. The principal directions of the graphite epoxy plies lie in the X1-X2 plane, where the angle is defined with respect to the X1 axes, see Fig. 

Q12. What is the buckling behavior of a graphite epoxy plate?

For the electrical loading an electric potential is applied to the upper and lower surface of the piezoelectric layers; where all three displacements of the middle surface at the boundary of the plate are fixed, see Fig. 3.According to [16] the material data are summarized in Tab. 

Q13. What is the orientation of the local basis system ti?

The orientation of the local basis system ti is defined by t1 in circumferential direction, t2 in axial direction and t3 in radial direction.