Isogeometric Analysis for Active Control of Piezoelectric Functionally Graded Plates in Thermal Environment
An isogeometric analysis (IGA) method is proposed for investigating the active shape and vibration control of functionally graded plates (FGPs) with surface-bonded piezoelectric materials in a thermal environment. A simple first-order shear deformation theory (S-FSDT) with four variables is used to describe the displacement field of the plates. To ensure the investigation of smart piezoelectric structure in the thermal environment closer to the actual situation, a modified piezoelectric constitutive equation with consideration of the temperature effect of dielectric and piezoelectric strain coefficients is implemented to replace the traditional linear piezoelectric constitutive equation. Meanwhile, the neutral surface is adopted to avoid the stretching-bending coupling. The accuracy and effectiveness of the proposed S-FSDT-based IGA method are verified by comparing with several existing numerical examples. Then, the static bending and open-loop control of the plates under mechanical and thermal loads are further studied. Finally, the active control including static bending control and vibration control of piezoelectric functionally graded plates (PFGPs) is also investigated by using a displacement-velocity feedback control law.
Smart piezoelectric structures are commonly used for active control in the fields of aerospace engineering . As a type of smart structures, piezoelectric functionally graded plates (PFGPs) refer to using the functionally graded plates (FGPs)  as the substrate and using the piezoelectric materials as sensors or actuators. In the last few decades, many scholars have studied the active control of FGPs with surface-bonded piezoelectric layers. For instance, He et al.  used the finite element method (FEM) to study the active vibration control of PFGPs and the voltage response of the actuators and sensors. Nguyen-Quang et al. [4, 5] analyzed the vibration control of FGPs and functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates. To avoid the reduction in calculation accuracy caused by element distortion , in FEM, Selim et al.  used the element-free method to research the active vibration control of functionally graded multilayer graphene nanoplatelets reinforced composite plates.
Indeed, the functional graded materials (FGMs) [8, 9] are often used in the thermal environment. Hence, the investigation of mechanical behavior of PFGPs in the thermal environment is conducive to improving the accuracy of structural analysis and design. Yang et al. [10, 11] estimated the nonlinear free vibration and static bending analysis of PFGPs in the thermal environment. Liew et al.  analyzed the static shape and vibration control of PFGPs. Dai et al.  researched the vibration control of PFGPs with the element-free method. However, they did not consider the temperature field of piezoelectric layers. An exact approach for nonlinear vibration and dynamic responses of PFGPs in the thermal environment was developed by Huang and Shen  and Xia and Shen . Similarly, Fakhari et al. [16, 17] developed a nonlinear investigation for the transient response and vibration control of PFGPs. Phung-Van et al.  employed isogeometric analysis (IGA)  to investigate the nonlinear dynamic response of PFGPs. Although the temperature field of piezoelectric materials was considered in [14–18], the effect of temperature on the properties of piezoelectric materials was not considered.
Different from these references, Shen  assumed that the material properties of PZT-5A vary linearly with temperature to research the thermal bending of PFGPs, but the effect of temperature on the dielectric parameter and piezoelectric strain parameter was not considered. The authors in  prove that the dielectric and piezoelectric strain parameters are sensitive to temperature. Undoubtedly, it is helpful to make the analysis closer to the actual situation if the influence of temperature on dielectric and piezoelectric strain parameters can be considered in the analysis of smart piezoelectric structure in the thermal environment.
For the numerical method of smart piezoelectric structures and nonhomogeneous composites, although FEM [22, 23] and meshless methods  have been proved to be successful and effective, some scholars are still trying to find superior numerical methods for research. Owing to the outstanding features of the exact geometrical modeling, high-order continuity, and simple meshing , IGA has been used in a variety of engineering applications [26–31]. Besides, the mathematical model of smart piezoelectric structures is complicated due to the existence of the electric field. Thus, an accurate method with low calculation cost is distinctly important. Obviously, the calculation efficiency and accuracy are directly affected by the number of variables in plate theories. In plate theories, classical plate theory (CPT)  has only three variables, whereas it is only appropriate for thin plates. There are five variables in first-order shear deformation theory (FSDT) , but the shear-locking effect will occur in the analysis of thin plates. Higher-order shear deformation theories (HSDTs)  and some other theories  have five or more variables, which make the discretization process more complicated. It is worth noting that, by adopting a simple first-order shear deformation theory (S-FSDT) with only four variables, Yu et al.  used the IGA method to investigate the free vibration and nonlinear static bending responses of FGPs. The results indicated that the S-FSDT-based IGA method is both applicable for thin and thick plates and naturally free from shear-locking.
This paper thus aims to develop an investigation for the active shape and vibration control of PFGPs in the thermal environment based on the S-FSDT and isogeometric analysis method. The neutral surface is introduced into the S-FSDT for avoiding the stretching-bending coupling. Additionally, to make the investigation of smart piezoelectric structure in the thermal environment close to the actual situation, a modified piezoelectric constitutive equation considering the temperature effect of dielectric and piezoelectric stress parameters is applied to substitute the traditional linear piezoelectric constitutive equation in the derivation of isogeometric finite element equation of PFGPs.
2. PFGPs in Thermal Environment
The sizes of the plate depicted in Figure 1 are (length × width × thickness). , where and are the thickness of FGP and piezoelectric layers. is the distance between the midsurface and neutral surface. and are the temperature of the top surface of PFGP and FGP. and are the temperature of the bottom surface of PFGP and FGP.
2.1. Material Properties
The material properties of the FGP are defined aswhere z is the thickness coordinate and . n is the gradient index. is the temperature. and represent the material properties of ceramic and metal.
In the thermal environment, and can be obtained as follows :where , , , , and are the material coefficients related to the temperature and are unique to the constituent materials.
2.2. Temperature Distribution
The temperature along the thickness can be acquired by calculating the steady-state heat transfer equation :where is the thermal conductivity.
Combined with the thermal boundary conditions and continuity conditions , the temperature distribution can be given by
2.3. Constitutive Equation
By introducing the influence of temperature on dielectric and piezoelectric stress parameters into the traditional linear piezoelectric constitutive equation , the modified piezoelectric constitutive equation is defined aswhere and are stress and strain. is the electric displacement vector. and are entropy and specific heat. is the electric field vector, and is the elastic constant matrix. , in which is the reference temperature. and are thermal expansion coefficient and pyroelectric vectors. and are temperature-dependent piezoelectric stress parameter and dielectric parameter matrices. The specific forms of , , , and arewithwhere and are piezoelectric stress and dielectric constant matrices at reference temperature. and are the coefficients that need to be confirmed through experiments.
2.4. Displacement Field and Strain
By introducing the neutral surface, the displacement field in S-FSDT  is given aswhere and denote the displacements of neutral surface. is given by
It is seen that the transverse displacement in FSDT [39, 40] is expressed by bending () and shear () terms; and the rotation variables are denoted by using the partial derivatives of the transverse bending components (, ) in S-FSDT. Through the above expressions, we can see that there are only four variables in S-FSDT.
The strain-displacement relation iswhere
The stress resultants of PFGPs can be calculated bywhere , , and denote in-plane forces, moments, and shear forces.
The thermal and electric stress resultants can be calculated bywith
2.5. Weak Form of Governing Equation
The equations of equilibrium and electrostatics can be denoted as follows :
where and are the components of the stress tensor and the body force. represents the component of the electric displacement vector. is the density.
By introducing the mechanical and electric boundary conditions [12, 30], the weak form of PFGPs can be obtained through Hamilton’s principle which can be expressed asin whichwhere and are the displacement and velocity. and represent mechanical point and surface loads, respectively. is the surface charge. and denote boundaries of corresponding to the external mechanical and the electrical loading surface, respectively.
2.6. Isogeometric Analysis
The NURBS basis function for the two-dimensional problems is expressed as follows :where and denote the B-spline basis functions with orders and in and directions, respectively. is the weight.
The displacements of the plate can be approximated aswhere represents the displacement vector of control point . denotes the NURBS basis function.
Similarly, the electric potential of piezoelectric layers can be approximated as
The electric field is then given by
The electric field in and directions is ignored; hence,
The specific form of items in equation (32) is given aswith
3. Closed-Loop Control
The active control of smart piezoelectric structures mainly includes two types: one is the open-loop control, which uses the inverse piezoelectric effect of piezoelectric materials to change the shape of the structures by applying voltage; the other is the closed-loop control, that is, using piezoelectric materials as sensors to sense the state of the structures; then, the output voltage of sensors is amplified as the input voltage of the actuators to control the deformation or suppress the vibration of structures.
In this paper, the upper piezoelectric layer is applied as an actuator, whereas the lower piezoelectric layer is used as a sensor. In general, the external charge of the sensor is zero. Therefore, using the subscripts and s to express the actuator and sensor, equation (31b) can be rewritten as
The electric potential generated in the sensor is given by
According to the displacement-velocity feedback control law , can be calculated bywhere and represent the displacement and velocity feedback control gains.
The Rayleigh damping in equation (42) can be calculated by
The Rayleigh damping coefficients and can be confirmed through .
4. Numerical Results
In this section, the convergence and accuracy of the S-FSDT-based IGA method is verified through some numerical examples. Then, the open-loop control and closed-loop control of piezoelectric functionally graded plates are investigated by using the modified piezoelectric constitutive equation. In the vibration control, the Newmark-β method  is adopted for researching the transient response of PFGPs, and the parameters γ and β are set to be 0.5 and 0.25, respectively.
4.1. Convergence Verification
Let us consider a simply supported PFGP with sizes of a = b = 400 mm, hf = 5 mm, and hp = 0.1 mm. The functionally graded plate is composed of ZrO2 and Al. The material parameters of metal and ceramic are reported in Table 1. The piezoelectric layers are PZT-G1195N, and the material properties are E11 = E22 = E33 = 63 GPa, , ρ = 7600 kg/m3, d31 = d32 = 254 × 10−12 m/V, and = 15 × 10−9 F/m.
Using the cubic NURBS basis functions, Table 2 lists the first six natural frequencies of the plate with different mesh levels. One can see that the results of this work are in good agreement with those in . Besides, the difference between the levels of 16 × 16 and 18 × 18 can be negligible. Therefore, the mesh of 16 × 16 is employed in the following analysis.
4.2. Comparison Studies
4.2.1. Piezoelectric Bimorph Beam
As shown in Figure 2, the beam consists of two PVDFs with opposite polarization directions. The sizes of the beam are 100 mm × 5 mm × 1 mm. The material properties are E11 = E22 = 2.0 GPa, G12 = 1.0 GPa, , e31 = e32 = 0.046 C/m2, and F/m. Figure 3 shows the centerline deflection of the beam under 1V voltage. One can see that the results in this work match well with the theory solution presented by Tzou .
4.2.2. Static Bending of FGP in Thermal Environment
The length and thickness of a square SSSS ZrO2/Al FGP are 200 mm and 10 mm. Assumed that the plate is subjected to the temperature gradient of and . is an increasing transverse distributed load from 0 to N/m2. Figure 4 plots the trend of the dimensionless central deflection () with the parameterized load (). Also based on the IGA, although the S-FSDT has only four variables, it is still as accurate and effective as HSDT .
4.3. Parametric Studies
4.3.1. Piezoelectric Stress and Dielectric Parameters
Similar to equations (8) and (9), we can also define the temperature-dependent piezoelectric stress parameter and dielectric parameter aswhere and are piezoelectric stress and dielectric constants at temperature .
Wang et al.  measured the piezoelectric strain constant and dielectric constant of PZT-5A at different temperature, where farad/m. Therefore, PZT-5A is selected as the actuators and sensors in subsequent investigations. Figure 5 plots the experimental data of and and their fitted curves. Through the equations of the fitted curves, and can be expressed as
4.3.2. Temperature Distribution of PFGP
We now study the temperature distribution of a PFGP with sizes of a = b = 400 mm, hf = 5 mm, and hp = 0.1 mm. The FGP is composed of ZrO2 and Al. The material properties of PZT-5H are E = 63 Gpa, , k = 2.1 W/mK, α = 5 × 10−9/°C, ρ = 7500 kg/m3, and P3 = 4 × 10−4 (C/m2°C).
Many studies [12, 13] of PFGPs ignored the heat conduction of the piezoelectric layers and directly applied thermal loads to the FGPs. Figure 6 shows the temperature distribution along the thickness of FGP under the temperature gradient of Tc = 100°C and Tm = 0°C. It is seen that the temperature through the thickness shows a linear distribution in the case that the plate is homogeneous. However, the temperature presents a nonlinear distribution and is lower than that in the homogeneous plate when the plate is non-homogeneous. Figure 7 shows the temperature distribution of PFGP under the temperature gradient of Tu = 100°C and Tl = 0°C. One can find that the temperature distribution of considering the heat conduction of the piezoelectric layers is completely different from that without considering the heat conduction. Besides, the temperature also presents a linear distribution when n is equal to zero. This is because the thermal conductivity of ZrO2 and PZT-5A are almost identical.
4.3.3. Static Bending of PFGPs
Figure 8 plots the centerline deflection of the plate subjected to a surface load of −100 N/m2. It is seen that the deflection increases with the increase in n. This is because the increase in n decreases the stiffness of the plate.
Figure 9 visualizes the centerline deflection under the thermal load Tc = 100°C and Tm = 0°C. It can be seen that all the deflections are downward, and the deflection of the plate with is larger than the deflection with n = 0. This is because the thermal expansion coefficient of Al is larger than ZrO2.
Figure 10(a) plots the centerline deflection under thermal load of Tu = 100°C and Tl = 0°C. The deflections are no longer all downward, and the deflection with n = 0 is greater than that with . This is on account of the thickness of the piezoelectric layer is very small, and the deflection is mainly generated by the temperature gradient of FGP. From Figure 7, we can see that the temperature distribution of the FGP with n = 0 is obviously larger than that of . Additionally, there is a difference between the deflection that considers the temperature effect of the piezoelectric parameters (in solid line) and the deflection that does not consider the temperature effect (in dash line). Figure 10(b) plots the centerline deflection with hp = 0.5 mm. One can observe that, with the increase in the thickness of the piezoelectric layers, the deformation of the plate is changed and the difference is amplified. Figure 11 shows the centerline deflection under the temperature gradient of Tu = 160°C and Tl = 0°C. The results indicate that, with the increase in temperature rise, the difference also increases because the modified piezoelectric constitutive method used in this paper is established according to the experimental data of the dielectric and stress parameters in the thermal environment. Certainly, it can help to calculate and simulate the mechanical behavior of smart piezoelectric structures closer to the actual situation.
Next, the open-loop control of PFGPs is studied, in which both two piezoelectric layers act as actuators. Figure 12 shows the centerline deflection of the plate with n = 0.5 under a mechanical load of −100 N/m2 and different voltages. Figure 13 plots the centerline deflection under the temperature gradient of Tu = 100°C and Tl = 0°C and different voltages. As expected, the deflection under mechanical and thermal loads can be reduced by increasing the input voltage.
4.3.4. Static Bending Closed-Loop Control
In the analysis of static bending closed-loop control, is used to control the static deformation of the plate, and is set to be zero. Figures 14 and 15 show the control effect of centerline deflection under the mechanical load of −100 N/m2 and temperature gradient of Tu = 100°C and Tl = 0°C, respectively. We can see that the deflection can be effectively controlled by using the displacement feedback control gain .
4.3.5. Active Vibration Control of PFGPs
The former CFFF PFGP is employed for investigating the active vibration control. Assumed that the plate suffers an initial transverse surface load of −100 N/m2, and then, the load is subsequently removed. Figures 16(a) and 16(b) are the transient responses at the tip of the plate under the temperature rise of Tu = 0°C, Tl = 0°C and Tu = 100°C, Tl = 0°C, respectively. It is seen that, as increases, the vibration disappears faster. Besides, because of the existence of thermal load, the vibration amplitude in Figure 16(b) is larger than that in Figure 16(a), and there is still a static offset after the vibration disappears in the thermal environment.
Finally, the active forced vibration control of a SSSS PFGP subjected to four types of distributed transverse loads is investigated. The index n = 1 and the sizes of plate are a = b = 200 mm, hf = 10 mm, and hp = 0.5 mm. Four types of loads are defined as follows:where N/m2 andin which and .
Figures 17–20 show the central deflection of the plate subjected to step, triangular, sinusoidal, and explosive loads. As expected, both in the forced and free vibration stages, the velocity feedback gain can control the vibration of the plate effectively. Because the deflection generated by temperature rise is opposite to that by mechanical loads, the deflection is negative after the vibration disappears in the thermal environment.
In this work, we employed a modified piezoelectric constitutive equation with consideration of the temperature effect of dielectric and piezoelectric strain parameters to replace the traditional linear piezoelectric constitutive equation for ensuring the investigation of smart piezoelectric structure in the thermal environment closer to the actual situation. In the selection of numerical methods, an isogeometric analysis (IGA) method combined with a simple first-order shear deformation theory (S-FSDT) was used. From some numerical examples, some meaningful conclusions can be obtained:(1)The temperature distribution of considering the heat conduction of piezoelectric layers is completely different from that without considering.(2)In thermal bending analysis of PFGPs, there is a difference between the results using the modified piezoelectric constitutive equation and those using the traditional linear piezoelectric constitutive equation. And this difference increases with the increase in temperature and thickness of piezoelectric layers.(3)The static deformation of the plates can be controlled according to the requirements by adjusting the control gain .(4)The free and forced vibration of the plates can be controlled effectively by using the velocity feedback control gain . When the plates are subjected to the thermal load, there is a static offset after the vibration disappears.(5)On the premise of ensuring the working performance of smart piezoelectric structures, to maintain the stability of piezoelectric structure in the thermal environment, the material whose piezoelectric parameters are not sensitive to temperature should be selected as far as possible.
Although the study in this paper has certain reference significance for the engineering application of smart piezoelectric structures, the NURBS-based IGA method still has some shortcomings , for instance, the waste of control points caused by refinement. Fortunately, these drawbacks can be avoided by using the rational splines over hierarchical T-meshes (RHT-splines)  and polynomial splines over hierarchical T-meshes (PHT-splines) . Thus, introducing these splines techniques into the present formulation would be an interesting work for our future studies.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the Anhui Science and Technology Major Project (Grant no. 201903a05020029), the University Synergy Innovation Program of Anhui Province (Grant no. GXXT-2019-048), the Open Project of Anhui Province Key Laboratory of Special and Heavy Load Robot (Grant no. TZJQR006-2021), and the Key Program of Natural Science Research of Higher Education in Anhui Province, China (Grant no. KJ2019A0091).
A. Menasria, K. Abdelhakim, A. A. Bousahla et al., “A four-unknown refined plate theory for dynamic analysis of FG-sandwich plates under various boundary conditions,” Steel and Composite Structures, vol. 36, pp. 355–367, 2020.View at: Google Scholar
K. Nguyen-Quang, H. Dang-Trung, V. Ho-Huu, H. Luong-Van, and T. Nguyen-Thoi, “Analysis and control of FGM plates integrated with piezoelectric sensors and actuators using cell-based smoothed discrete shear gap method (CS-DSG3),” Composite Structures, vol. 165, pp. 115–129, 2017.View at: Publisher Site | Google Scholar
K. Nguyen-Quang, T. Vo-Duy, H. Dang-Trung, and T. Nguyen-Thoi, “An isogeometric approach for dynamic response of laminated FG-CNT reinforced composite plates integrated with piezoelectric layers,” Computer Methods in Applied Mechanics and Engineering, vol. 332, pp. 25–46, 2018.View at: Publisher Site | Google Scholar
A. Boussoula, B. Boucham, M. Bourada, F. Bourada et al., “A simple nth-order shear deformation theory for thermomechanical bending analysis of different configurations of FG sandwich plates,” Smart Structures and Systems, vol. 25, pp. 197–218, 2020.View at: Google Scholar
S. Refrafi, A. A Bousahla, A. Bourada et al., “Effects of hygro-thermo-mechanical conditions on the buckling of FG sandwich plates resting on elastic foundations,” Computers and Concrete, vol. 25, pp. 311–325, 2020.View at: Google Scholar
J. Yang, S. Kitipornchai, and K. M. Liew, “Non-linear analysis of the thermo-electro-mechanical behaviour of shear deformable FGM plates with piezoelectric actuators,” International Journal for Numerical Methods in Engineering, vol. 59, no. 12, pp. 1605–1632, 2004.View at: Publisher Site | Google Scholar
K. M. Liew, X. Q. He, T. Y. Ng, and S. Sivashanker, “Active control of FGM plates subjected to a temperature gradient: modelling via finite element method based on FSDT,” International Journal for Numerical Methods in Engineering, vol. 52, no. 11, pp. 1253–1271, 2001.View at: Publisher Site | Google Scholar
X. L. Huang and H. S. Shen, “Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments,” Journal of Sound and Vibration, vol. 289, pp. 25–53, 2005.View at: Google Scholar
V. Fakhari and A. Ohadi, “Nonlinear vibration control of functionally graded plate with piezoelectric layers in thermal environment,” Journal of Vibration and Control, vol. 17, no. 3, pp. 449–469, 2009.View at: Google Scholar
P. Phung-Van, L. V. Tran, A. J. M. Ferreira et al., “Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads,” Nonlinear Dynamics, vol. 87, no. 2, pp. 1–16, 2016.View at: Publisher Site | Google Scholar
H. S. Shen, “Nonlinear thermal bending response of FGM plates due to heat conduction,” Composites Part. B: Engineering, vol. 38, no. 2, pp. 201–215, 2006.View at: Google Scholar
S. Alimirzaei, M. Mohammadimehr, and A. Tounsi, “Nonlinear analysis of viscoelastic micro-composite beam with geometrical imperfection using FEM: MSGT electro-magneto-elastic bending, buckling and vibration solutions,” Structural Engineering and Mechanics, vol. 71, no. 5, pp. 485–502, 2019.View at: Google Scholar
M. S. Kiani and M. R. Eslam, “Dynamic analysis and active control of smart doubly curved FGM panels,” Composite Structures, vol. 102, pp. 205–216, 2013.View at: Google Scholar
N. Bendenia, M. Zidour, A. A. Bousahla et al., “Deflections, stresses and free vibration studies of FG-CNT reinforced sandwich plates resting on Pasternak elastic foundation,” Computers and Concrete, vol. 26, no. 3, pp. 213–226, 2020.View at: Google Scholar
A. Bakora, F. Bourada, A. A. Bousahla et al., “Buckling analysis of functionally graded plates using HSDT in conjuction with the stress function method,” Computers and Concrete, vol. 27, no. 1, 2021.View at: Google Scholar
T. T. Yu, S. Yin, T. Q. Bui et al., “A simple FSDT-based isogeometric analysis for geometrically nonlinear analysis of functionally graded plates,” Finite Elements in Analysis and Design, vol. 96, pp. 1–10, 2014.View at: Google Scholar
A. Lal, N. L. Shegokar, and B. N. Singh, “Finite element based nonlinear dynamic response of elastically supported piezoelectric functionally graded beam subjected to moving load in thermal environment with random system properties,” Applied Mathematical Modelling, vol. 44, pp. 274–295, 2017.View at: Publisher Site | Google Scholar
N. V. Nguyen, L. B. Nguyen, H. Nguyen-Xuan et al., “Analysis and active control of geometrically nonlinear responses of smart FG porous plates with graphene nanoplatelets reinforcement based on Bézier extraction of NURBS,” International Journal of Mechanical Sciences, vol. 180, 2020.View at: Google Scholar
M. M. Al-Zahran, A. Rouabhia, C. Abdelbaki et al., “Physical stability response of a SLGS resting on viscoelastic medium using nonlocal integral first order theory,” Steel and Composite Structures, vol. 37, no. 6, pp. 695–709, 2020.View at: Google Scholar
A. A. Bousahla, F. Bourada, S. R. Mahmoud et al., “Buckling and dynamic behavior of the simply supported CNT-RC beams using an integral-first shear deformation theory,” Computers and Concrete, vol. 25, no. 2, pp. 155–166, 2020.View at: Google Scholar
C. Wang, J. M. Koh, T. T. Yu et al., “Material and shape optimization of bi-directional functionally graded plates by GIGA and an improved multi-objective particle swarm optimization algorithm,” Computer Methods in Applied Mechanics Engineering, vol. 366, 2020.View at: Publisher Site | Google Scholar
H. Tzou, “Development of a light-weight robot end-effector using polymeric piezoelectric bimorph,” in Proceedings of the IEEE. International Conference on Robotics and Automation, Scottsdale, AL, USA, May 1989.View at: Google Scholar
N. Nguyen-Thanh, H. Nguyen-Xuan, S. P. A. Bordas, and T. Rabczuk, “Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 21-22, pp. 1892–1908, 2011.View at: Publisher Site | Google Scholar