The numerical analysis of piezoelectric ceramics based on the Hermite-type RPIM
Introduction
Piezoelectric ceramics have been used widely in sensors and actuators owing to its excellent performance, and it has played an important role in micro electro-mechanical systems and non-destructive testing.
In the piezoelectric ceramics, a voltage will be induced in the material when subjected to a mechanical deformation, which is known as the direct piezoelectric effect. Similarly, a strain will be generated if a voltage is applied across the material, which is known as the indirect piezoelectric effect. A lots of instruments have been designed based on the direct and indirect piezoelectric effect, such as pressure sensors [1], strain gauges, microphones, ultrasonic motors, accelerometers [2]; piezoelectric bender elements, linear extension elements, ultrasonic rotary motors [3].
In over half a century, the finite element method (FEM) has become a useful method to solve the engineering problems including complex structures and non-linear problems [4], [5]. Besides, the FEM has been successfully used in the electromechanical coupling partial differential equations of the piezoelectric ceramics. Many studies have shown that the FEM can meet the requirements of accuracy and stability of solving the piezoelectric ceramics problems in engineering disciplines. However, the FEM needs much time and work to achieve the refinement of meshes for the situation, in which the local regions should be solved accurately for the complex structures.
Thus, the meshless methods have been put forward to overcome the problems mentioned above in the FEM [6], [7], [8], [9]. The distinction between the meshless methods and the FEM is that the meshless methods use a series of appropriate scattered nodes to express the problem domain and boundary, and construct the interpolation function to approximate the problem domain. The relation of connection among nodes is not necessary for the meshless methods. If we need to solve the local regions of problem domain accurately, only the number of the nodes should be increased. Clearly, it is easier to achieve, compared with the FEM.
In recent 20 years, the meshless methods have developed quickly. They mainly consist of the reproducing kernel particle method [10], [11], [12], the radial basis function method [13], [14], the element-free Galerkin method [15], [16], [17], [18], the finite point method [19], [20], the partition of unity method [21], [22], the polynomial point interpolation method [23], [24], the moving least-squares method [25], [26], [27], the local Petrov−Galerkin method [28], [29], [30], [31], the smooth particle hydrodynamics [32], [33], the boundary integral equation method [34], [35], Hermite radial point interpolation method [36], [37], [38], and meshless manifold method [39], [40].
Compared with the existing meshless methods, the Hermite-type RPIM has its own specific features: fast speed and high precision, so the piezoelectric ceramics can be analyzed by applying the Hermite-type RPIM. In this paper, the inside and boundary of the problem domain is discreted by a series of nodes, and the approximate displacement function of evaluation nodes are constructed by using the Hermite-type RPIM. The two corrected coefficients are introduced to make approximate displacement function close to exact displacement function, and the displacement of evaluation nodes are solved by the approximate displacement function. Additional nodes can be simply added to regions, if more accuracy is required. Finally, the examples are presented to demonstrate the accuracy and stability of the Hermite-type RPIM.
Section snippets
The governing equations of piezoelectric ceramics
In the x−z plane, the two-dimensional piezoelectric ceramics constitutive equations can be expressed in two aspects of the stress and the electric field. where ε, σ,Ek and Di are the strain tensor, the stress tensor, the electric field vector and the electric displacement vector, respectively. e,cE and ξεare the piezoelectric constant, elastic stiffness, and dielectric constant, respectively. Superscript ε and E represent coefficients measured at constant electric
The approximate displacement function of piezoelectric ceramics constructed by the Hermite-type RPIM
In the z-direction, the approximate displacement function wh(x,z) can be expressed as the linear combination of RBFs (Racial Basis Function, constructed by n nodes), normal derivative constructed by nodes of DBs (nodes on the interpolation function derivative condition) and m polynomial functions in local supporting domain, as Fig. 1 where ai, bj and ck are undetermined coefficients; n is the number of nodes in the local supporting
Correction of approximate function
Substituting Eq.(36) into Eq.(35), the detailed expression of the approximate function can be written as
In order to improve the accuracy of Eq.(37), the two corrected coefficients of μ, η are introduced to make approximate displacement function close to exact displacement function. Eq.(37) can be rewritten as the form of the interpolation function and its derivative
Mechanical deformation of the bimorph with 1 V
A 10 × 1 µm(L = 10 µm, 2 h = 1 µm) bimorph made of PVDF is taken as an example to analyze the displacements, as Fig 2. The bimorph is divided into the top and bottom layers, and each layer is represented by nodes. Nodes lying on the interface are placed to coincide with the adjoining nodes on the other side. After discreting the problem domain by nodes, the approximate function are constructed by applying the Hermite-type RPIM.
The following boundary conditions apply to the top layer
Conclusions
In this paper, the characteristic of the piezoelectric ceramics is indicated by the electromechanical coupling partial differential equations, then the partial differential equations can be solved by the Hermite-type RPIM. This method uses nodes to discrete the inside and boundary of the problem domain instead of meshing, and constructs approximate displacement function. The two corrected coefficients are introduced to make approximate displacement function close to exact displacement function,
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant number 11271234).
References (40)
- et al.
A square-type rotary ultrasonic motor using longitudinal modes
J. Electroceramics
(2014) Particle method for hydrodynamics
Comput. Phys. Rep.
(1985)- et al.
Error estimates for the interpolating moving least-squares method in n-dimensional space
Appl. Numer. Math.
(2015) - et al.
Numerical solution of transient heat conduction problems using improved meshless local Petrov−Galerkin method
Appl. Math. Comput.
(2013) - et al.
A novel capacitive pressure sensor and interface circuitry
Microsyst. Technol.
(2013) - et al.
Agreement in regard to total sleep time during a nap obtained via a sleep polygraph and accelerometer: a comparison of different sensitivity thresholds of the accelerometer
Int. J. Behav. Med.
(2012) - et al.
Fully discrete mixed finite element method for a quasilinear stefan problem with a forcing term in non-divergence form
J. Appl. Math. Comput.
(2007) A splitting positive definite mixed finite element method for two classes of integro-differential equations
J. Appl. Math. Comput.
(2012)- et al.
Smoothed particle hydrodynamics: theory and applications to non-spherical stars
Mon. Not. R. Astron. Soc.
(1977) - et al.
Combining the complex variable reproducing kernel particle method and the finite element method for solving transient heat conduction problems
Chin. Phys. B
(2013)