A New BEM Modeling Algorithm for Size-Dependent Thermopiezoelectric Problems in Smart Nanostructures

The main objective of this paper is to introduce a new theory called size-dependent thermopiezoelectricity for smart nanostructures. The proposed theory includes the combination of thermoelastic and piezoelectric influences which enable us to describe the deformation andmechanical behaviors of smart nanostructures subjected to thermal, and piezoelectric loadings. Because of difficulty of experimental research problems associated with the proposed theory. Therefore, we propose a new boundary element method (BEM) formulation and algorithm for the solution of such problems, which involve temperatures, normal heat fluxes, displacements, couple-tractions, rotations, force-tractions, electric displacement, and normal electric displacement as primary variables within the BEM formulation. The computational performance of the proposed methodology has been demonstrated by using the generalized modified shift-splitting (GMSS) iteration method to solve the linear systems resulting from the BEM discretization. GMSS advantages are investigated and compared with other iterative methods. The numerical results are depicted graphically to show the size-dependent effects of thermopiezoelectricity, thermoelasticity, piezoelectricity, and elasticity theories of nanostructures. The numerical results also show the effects of the sizedependent and piezoelectric on the displacement components. The validity, efficiency and accuracy of the proposed BEM formulation and algorithm have been demonstrated. The findings of the current study contribute to the further development of technological and industrial applications of smart nanostructures.

manufacturing devices that can be used to study the properties of nanomaterials [1,2]. Nanostructures are one of the main products of nanotechnology. A nanostructure is a structure that has at least one dimension equal or less than 100 nanometers. Understanding the mechanical behaviour of deformed nanostructures is of great importance due to their applications in all fields such as industry, medicine, renewable energy, military and civil and architecture Engineering. In the field of industry, certain nanoparticles can be used in the manufacture of filters to purify and desalinate water more efficiently than other types of filters, and they are also used as a heat insulator with high efficiency. Some nanomaterials such as tungsten carbide and silicon carbide are distinguished by their high strength compared to ordinary materials, so they are used in the manufacture of some tools Cutting and drilling. Dust and water-repellent paints, clothing, and glass can also be made [3]. Recent developments in nanoscale electronics and photonics might lead to new applications such as high-density memory, high-speed transistors and high-resolution lithography [4][5][6]. In the medical field, certain nanoparticles can be used as drug-carrying materials, as these materials have a special sensitivity to the place to which the drug is intended to be sent, so when they reach it inside the human body, they release the drug accurately, in addition to promising research confirming the possibility of using nanomaterials as a treatment for cancer. Gold nanoparticles are also used in home testing devices to detect pregnancy. Nanowires are used as nanoscale biosensors to detect a large number of diseases in their early stages [7,8]. In the field of renewable energy, nanomaterials are involved in the manufacture of solar cells that are used in the production of electrical energy, where materials such as cobalt oxide or semi-conductive materials in general such as silicon and germanium are deposited on glass sheets or silica plates and because these materials have a nanoscale size, the surface area that is exposed to sunlight is greater, and thus we ensure that we absorb the largest amount of sunlight in a single cell. The panel usually consists of hundreds of solar cells that are connected through an electrical circuit that converts solar energy into electrical energy. In the military field, nanomaterials enter into the manufacture of nanoscale cylinders that are characterized by strength and rigidity, in addition to a storage capacity a million times greater than regular computers, the manufacture of military clothing that has the ability to absorb radar waves in order to stealth and infiltrate, and the manufacture of nanosatellites [9][10][11]. In the field of building and construction, some nanomaterials such as titanium dioxide TiO2, carbon nanotubes CNTs and silica nanoparticles are added to concrete to increase the durability and hardness of the concrete in addition to increasing its resistance to water penetration. Size-dependent porothermoelastic [12][13][14][15] interactions play a significant role in many areas of nanotechnology applications. Because of computational complexity in solving sizedependent thermopiezoelectric problems not having any general analytical solution [16], therefore, numerical methods should be developed to solve such problems. Among these numerical methods is the boundary element method (BEM) that has been used for engineering models [17], bioheat transfer models [18], and nanostructures [19]. The main feature of BEM [20] over the domain type methods [21] is that only boundary of the considered domain needs to be discretized. This feature is of great importance for solving complex nanoscience and nanotechnology problems with fewer elements, and requires less computational cost, less preparation of input data, and therefore easier to use.
In the present paper, we introduce a new theory called size-dependent thermopiezoelectricity for smart nanostructures to describe the mechanical behaviors of deformed nanostructures subjected to various types of mechanical, thermal, and piezoelectric loadings. Also, we develop a new boundary element formulation for solving the deformation problems associated with the proposed theory. The numerical results illustrate the size-dependent effects on the thermo-piezoelectric, thermoelastic, piezoelectric, and elastic smart nanostructures. The numerical results also show the effects of the length scale parameter and piezoelectric coefficient on the displacement components, and confirm the validity, efficiency and accuracy of the proposed BEM formulation and algorithm.

Formulation of the Problem
Consider a size-dependent thermopiezoelectric nanostructure occupies the cylindrical region V (cross section of the nanostructure in the x 1 x 2 -plane) that bounded by S, such that x3-axis parallel to the cylinder axis, as shown in Fig. 1. We take n α to be the outward unit vector that is perpendicular to the boundary surface S as follows where ε αβ (ε 12 = −ε 21 = 1, ε 11 = ε 22 = 0) is the two-dimensional permutation symbol. In the two-dimensional plane, all quantities are independent of x 3 . The deformation is described by the displacement vector u = (u 1 , u 2 ), and the electric effect is specified by the electric potential φ The rotation component is where The electric field components are The strain tensor and the mean curvature vector where The true couple-stress vector M i can be expressed in terms of pseudo couple-stress tensor M kj as where the true couple-stress vector The force-stress tensor can be decomposed into the following two parts where σ αβ (σ 3α = σ α3 = 0) is the force-stress tensor, σ (αβ) is the symmetric force-stress tensor and σ [αβ] is the skew-symmetric force-stress tensor.
The electric field and mechanical deformation can induce polarization P α in the piezoelectric material. The electric displacement D α is given as where ε 0 is the vacuum permittivity, E α is the electric field, P α is the polarization of piezoelectric material.
The governing equations of size-dependent thermopiezoelectric problems in smart nanostructures subjected to various types of mechanical, thermal and piezoelectric loadings can be expressed as The entropy balance equation where q α is the heat flux vector.
The force equilibrium equation where F α is the body force vector.
The moment equilibrium equation [21] = −σ [12] = −M [1,2] The Gauss's law for electric field can be expressed as where ρ E is the volume electric charge density.
Substitution of Eqs. (12) and (17) into force equilibrium Eq. (16) leads to The constitutive relations of size-dependent thermopiezoelectric nanostructures can be written as: The heat flux vector equation The symmetric force-stress equation (15) where α is the coefficient of thermal expansion, δ αβ is the Kronecker delta function the couple-stress equation the electric displacement equation Also, the force-traction vector t α , couple-traction m, and normal electric displacement d can be written as follows where f is the piezoelectric coefficient.
The Lamé elastic constants λ ad μ for an isotropic material, can be related to the Poisson ratio v and Young's modulus E as where v is the Poisson ratio, E is the Young's modulus, The electric permittivity of the material can be defined as where ε r is relative permittivity.
The material length scale parameter used in couple stress theories can be written as where η is the couple-stress parameter.
Now, the total force-stress tensor σ βα can be expressed as Hence, the governing Eqs. (9), (10) and (12) can be written as where k is the thermal conductivity, T is the temperature and Q is an external heat source.
Now, the normal heat flux q, force-traction vector t α , couple-traction m, and normal electric displacement d can be written as follows

Boundary Conditions
The considered boundary conditions may specify either temperature change T or normal heat flux q Displacements u α or force-tractions t α Rotation ω or couple-traction m and electric potential φ or normal electric displacement d where S T , S q , S u , S t , S ω , S m , S φ and S d are the boundary parts at which the boundary values for the temperature change T, the normal heat flux q, the displacement vector u α , the force-traction vector t α , the rotation ω, the couple-traction m, the electric potential φ and the normal electric displacement d are specified.

Boundary Element Implementation
Now, we can write the boundary integral equations for temperature, displacements, rotation, and potential as follows where the superscripts Q * , F * , C * and R * are chosen to be kernel functions associated with point heat source, point force, point couple and point electrical source infinite space fundamental solutions, respectively, and denotes the Cauchy principal value symbol. denotes the Cauchy principal value symbol. The full details for the derivations of the fundamental solutions used in the current formulation are given in [22][23][24].

The integral Eqs. (40)-(43) in absence of body forces and volume charge density can be written in matrix form as follows
Now, it is convenient to rewrite Eq. (44) in compact index-notation form as where the generalized displacements u I in (45) include temperature T, displacement u α , rotation ω and electric potential φ, respectively. Similarly, the generalized tractions t I include normal flux q, force-tractions t α , couple-traction m and normal electric displacement d, respectively.
This leads to the following linear algebraic equations system where T and U are dense matrices related with the left and right hand sides of Eq. (44), respectively, and u represents the nodal boundary temperature, displacement, rotation and electric potential, respectively, while, t represents the nodal boundary normal flux, force-tractions, couple-tractions and normal electric displacement, respectively.
which can be written as where A is the non-symmetric dense matrix, B is the known boundary values vector and X is the unknown boundary vector of unknown boundary values vector.

Numerical Results and Discussion
To illustrate the numerical calculations computed by the proposed methodology, we consider the thermopiezoelectric nanoplate with free boundary conditions on the sides, as shown in Fig. 2. A variable temperature field in the x 2 -direction is generated by applying T b and T t to the bottom and top surfaces, respectively. Also, a uniform electric field in the x 2 -direction is generated by applying constant electric potentials φ b and φ t to the bottom and top surfaces, respectively. Under thermal and piezoelectric loadings, the plate deforms and becomes electrically polarized. As a result, the thermal effect is specified by the thermal expansion coefficient α, the size dependent effect is specified by one characteristic length scale parameter l, and the piezoelectric effect is specified by one piezoelectric coefficient f . The solid line represents the Case A that corresponds to the size-dependent thermopiezoelectric plates (α = 1, f = −1) . The dashed line represents the Case B which corresponds to size-dependent thermoelastic plates (α = 1, f = 0). The dash-dot line represents the Case C that corresponds to size-dependent piezoelectric plates (α = 0, f = −1). The dash-two dot line represents the Case D which corresponds to size-dependent elastic plates (α = 0, f = 0).

Figs. 3 and 4
show the variation of the displacements u 1 and u 2 along x-axis for different size-dependent theories. It can be seen from these figures that the differences between sizedependent thermopiezoelectricity, size-dependent thermoelasticity, size-dependent piezoelectricity, and size-dependent elasticity theories are very pronounced.   The efficiency of our proposed methodology has been demonstrated through the use of the GMSS iteration method [25], which reduces the memory requirements and Processing time [26,27]. During our treatment of the considered problem, we implemented symmetric successive over relaxation (SSOR) [28], and preconditioned generalized shift-splitting (PGSS) iteration methods [29] to solve the linear systems resulting from the BEM discretization. Tab. 1 illustrates the iterations number (Iter.), processor time (CPU time), relative residual (Rr), and error (Err.) of the considered methods computed for various length scale parameter values (l = 0.01, 0.1 and 1.0). It is shown from Tab. 1 that the GMSS needs the lowest IT and CPU times, which implies that GMSS method has better performance than SSOR and PGSS.     -Because of the benefits of the BEM such as dealing with more complicated shapes of nanostructures and not requiring the discretization of the internal domain, also, it has low CPU time and memory. Therefore, it is versatile and efficient method for modeling of size-dependent thermopiezoelectric problems in smart nanostructures.
-A new BEM formulation is developed for solving the problems associated with the proposed theory, which involves temperatures, normal heat fluxes, displacements, couple-tractions, rotations, force-tractions, electric displacement, and normal electric displacement as primary variables within the BEM formulation.
-The BEM is accelerated by using the GMSS, which reduces the total CPU time and number of iterations.
-The proposed theory includes the combination of thermoelastic and piezoelectric influences which enable us to explain the differences between size-dependent thermopiezoelectricity, sizedependent thermoelasticity, size-dependent piezoelectricity and size-dependent elasticity theories of nanostructures.
-Numerical findings are presented graphically to show the effects of the size-dependent and piezoelectric on the displacement components.
-The computational performance of the proposed methodology has been demonstrated. -From the proposed model that has been carried out using BEM formulation, it is possible to conclude that our proposed technique is more convenient, cost-effective, highly accurate, and has superiority over FDM or FEM.
-The proposed technique can be applied to study a wide variety of size-dependent problems in smart nanostructures subjected to mechanical, thermal and piezoelectric loadings.
-It can be concluded that our study has a wide variety of applications in numerous fields, such as electronics, chemistry, physics, biology, material science, optics, photonics, industry, military, and even medicine.
-Current numerical results for the proposed theory and its related problems, may provide interesting information for nanophysicists, nanochemists, nanobiologists, nanotechnology engineers, and nanoscience mathematicians as well as for computer scientists specializing in nanotechnology.

Funding Statement:
The author received no specific funding for this study.

Conflicts of Interest:
The author declares that he has no conflicts of interest to report regarding the present study.