The main aim of this paper is to propose a new memory dependent derivative (MDD) theory which called three-temperature nonlinear generalized anisotropic micropolar-thermoelasticity. The system of governing equations of the problems associated with the proposed theory is extremely difficult or impossible to solve analytically due to nonlinearity, MDD diffusion, multi-variable nature, multi-stage processing and anisotropic properties of the considered material. Therefore, we propose a novel boundary element method (BEM) formulation for modeling and simulation of such system. The computational performance of the proposed technique has been investigated. The numerical results illustrate the effects of time delays and kernel functions on the nonlinear three-temperature and nonlinear displacement components. The numerical results also demonstrate the validity, efficiency and accuracy of the proposed methodology. The findings and solutions of this study contribute to the further development of industrial applications and devices typically include micropolar-thermoelastic materials.

The study of thermoelastic models has recently gained growing attention due to its many applications in aerospace technologies, geophysics, aeronautics, astronautics, robotics, earthquake engineering, mining engineering, nuclear energy industry, military technologies, soil dynamics, high-energy particle accelerators and detectors, and other engineering and electronic industries [

The classical thermo-elasticity (CTE) theory of Duhamel [

The fractional calculus is the mathematical branch that is used to study the theory and applications of derivatives and integrals of arbitrary non-integer order. Recently, this branch has emerged as an effective tool for modeling of various engineering and industrial applications [

Several famous mathematicians have contributed to the development of fractional order calculus, where Euler mentioned interpolating between integral orders of a derivative in 1730. At that point, Laplace characterized a fractional derivative by implies of an integral in 1812.

Lacroix presented the first formula for the fractional order derivative appeared in 1819, where he introduced the

In 1967, the Italian mathematician Caputo presented his fractional derivative of order

Diethelm [

where

Wang et al. [

where the first order (

Based on several practical applications, the memory effect needs weight

As a special case

The above equation shows that the common derivative

Due to the computational difficulties in solving nonlinear generalized anisotropic thermoelastic problems, the problems become too complicated with no general analytical solution. So, numerical solutions should be implemented instead of analytical solutions to obtain the approximate solutions for such problems, one of the best of these numerical methods is the boundary element method (BEM) [

Researchers in numerical methods were only aware of the importance of FEM which could solve complex engineering problems. But now after the huge achievements of BEM and their ability to solve inhomogeneous and non-linear problems involving infinite and semi-infinite domains very efficiently, they realized the power, ease and accuracy of BEM in solving their complex problems by using a lot of software like FastBEM and BEASY.

The main aim of this paper is to propose a new MDD theory, called three-temperature nonlinear generalized anisotropic micropolar-thermoelasticity and propose a novel BEM technique for solving problems associated with the proposed theory. The numerical findings are graphically represented to demonstrate the impacts of the time delays and kernel functions on the total nonlinear three-temperature and nonlinear displacement components and demonstrate the validity and exactness of the suggested technique.

A brief summary of this paper is as follows: Section 1 introduces the background and provides the readers with the necessary information to books and articles for a better understanding of thermoelasticity theories, memory dependent derivative history and their applications. Section 2 describes the physical modeling of memory dependent derivative problems of three-temperature nonlinear generalized anisotropic micropolar-thermoelasticity. Section 3 outlines the BEM implementation for obtaining the temperature field of the considered problem. Section 4 outlines the BEM implementation for obtaining the displacement field of the considered problem. Section 5 introduces the computational performance of the proposed technique. Section 6 presents the new numerical results that describe the effects of time delays and kernel functions on the total temperature and displacement components. Section 7 outlines the significant findings of this paper.

The geometry of the considered problem is shown in _{i}_{1} +_{2} = _{3} +_{4} =

The memory dependent derivative governing equations for three-temperature nonlinear generalized anisotropic micropolar-thermoelasticity theory and its problems can be expressed as follows [

where

The two dimensions three temperature (2D-3T) radiative heat conduction equations can be expressed as

where _{ij}_{k}_{abfg}_{abfg}_{fgab}_{bafg}_{i}_{i}_{1}, _{2},

The 2D-3T radiative heat conduction equations mentioned above

where

where

and

which can be written in memory dependent derivative form as follows

where

and

where

Initial and boundary conditions of 3T field can be written as

By using the fundamental solutions

Now, by applying the technique of Fahmy [

which, in the absence of heat sources, can be written as follows

In order to transform the domain integral of

where ^{j}(

We assume that

Then,

where

and

where

By discretizing

where

The diffusion matrix may be described as

where

To solve

where,

By differentiating

By substituting from

which can be written as follows

in which

Use of the weighted residual method to the governing

where

in which _{i}

The boundary conditions are

By integrating the first term of

Based on Huang et al. [

By integrating the left-hand side of

Based on Eringen [

Thus,

By integrating the left-hand side of _{i}_{i}

The weighting functions for _{i}_{l}

Now, we consider the following analytic fundamental solution of Dragos [

The weighting functions for _{i}_{l}

The analytic fundamental solution of Dragos [

By using the above weighting functions sets into

Thus, we can write

where

In order to solve

By substituting from

which also can be written as

By applying the following definition

Thus, by using

Hence, the global matrix system can be expressed as

Now, by using the initial and boundary conditions, we can write

in which

Nowadays, modern CPUs are very powerful, versatile and can perform very complex problems much faster than previous ones [

The main objective of this section is to implement an accurate and robust iteration technique for solving the dense nonsymmetric algebraic system of linear equations arising from the BEM. So, GMSS of Huang et al. [

Method | Parameters | IT | CPU | RES | ERR | |
---|---|---|---|---|---|---|

0.003 | GMSS | 40 | 0.0513 | 6.36e-07 | 4.92e-08 | |

Regularized | 10 | 70 | 0.0948 | 9.25e-07 | 5.48e-07 | |

Uzawa-HSS | 80 | 0.1120 | 9.82e-07 | 5.98e-06 | ||

0.03 | GMSS | 45 | 0.0942 | 2.46e-06 | 3.24e-07 | |

Regularized | 10 | 110 | 0.4231 | 3.25e-05 | 5.89e-06 | |

Uzawa-HSS | 120 | 0.5102 | 7.32e-05 | 3.46e-05 | ||

0.3 | GMSS | 49 | 0.1046 | 5.23e-05 | 2.45e-06 | |

Regularized | 10 | 250 | 0.8973 | 4.89e-04 | 4.87e-05 | |

Uzawa-HSS | 300 | 0.9782 | 1.09e-03 | 7.84e-04 |

Now, the resulted linear system

where

The iteration scheme of Uzawa method can be defined as

Due to the effectiveness of Uzawa method, several generalized techniques of Uzawa method, such as parameterized Uzawa methods, preconditioned Uzawa methods, inexact Uzawa methods and parameterized inexact Uzawa methods, have been developed to solve

In order to solve the linear system

In order to describe the Uzawa-HSS, we consider the iteration scheme of HSS iteration method which is used for solving linear equations system

which equals to

where

Now, we can define the Uzawa–HSS iteration scheme as follows:

First, compute

Second, compute

where

For

where

Now, the resulted linear system

where

According to Cao et al. [

Now, the iteration scheme of the modified shift-splitting (MSS) can be described for solving linear equations system

Based on the MSS iteration method, the generalized modified shift-splitting (GMSS) for the nonsymmetric matrix

For

where

The GMSS iteration method can be expressed as

where

and

According to Huang et al. [

Let

Now, the GMSS iteration method can be derived using the following algorithm:

For a given vector

Compute

Solve

Compute

It can be seen from algorithm 1 that a linear system with the coefficient matrix

Badahmane [

where

According to Badahmane [

For

The GMSS iteration method can be expressed as

where

where the regularized preconditioner of the matrix

From

which equals to

where

At each iteration step of regularized iteration

Solve

Compute

The technique proposed in the current study may be applicable to a wide variety of three-temperature micropolar-thermoelastic problems relating to the suggested theory. During the simulation process the effects of time-delay and kernel function play a very important role. The proposed technique has been proven to be successful and efficient.

In the considered boundary element model, the boundary has been discretized using 42 linear boundary elements and 68 internal points as shown in

_{1} and _{2} along

_{1} and _{2} along

As there are no findings available for the problem under consideration. So, some literatures may be regarded as special cases from our general BEM problem. For comparison purposes with other approaches special cases addressed by other authors, we considered only one-dimensional problem. In the special case under consideration, the results are plotted in

The performance of GMSS iteration method is compared against Uzawa-HSS iteration method and regularized iteration method. In actual computation, the parameters,

The main purpose of the current paper is to propose a new MDD theory called three-temperature nonlinear generalized anisotropic micropolar-thermoelasticity. This theory forms a new and good research point in thermoelasticity, and the scientific community will be interested in studying this research point in the following years due to its numerous low-temperature and high-temperature applications. The problems related to the proposed theory are very difficult to solve analytically. Therefore, we propose a new boundary element technique for solving such problems. For comparison purposes with other researchers in the literature, we only considered the one-dimensional one-temperature heat conduction model as a special case of our three-temperature heat conduction model. The numerical results confirm the validity and exactness of our suggested technique, where the BEM results are in excellent agreement with the results of FDM and FEM.

The GMSS iteration method has been implemented for solving the resulting linear systems in order to reduce the iterations number and CPU time. The implemented GMSS iteration method is quickly convergent without needing complicated calculations and. On the other hand, it is anticipated that the GMSS iteration method with the optimal parameters

The numerical results of our considered study can provide data references for mechanical engineers, computer engineers, geotechnical engineers, geothermal engineers, technologists, new materials designers, physicists, material science researchers and those who are interested in novel technologies in the area of three-temperature micropolar generalized thermoelastic materials. Application of three-temperature theories in advanced manufacturing technologies, with the development of soft machines and robotics in biomedical engineering and advanced manufacturing, thermoelastic response will be encountered more often where three-temperature radiative heat conduction will turn out to be the best choice for thermomechanical analysis in the design and analysis of micropolar generalized thermoelastic materials and structures.

The authors would like to thank the anonymous reviewers and the editor for their useful suggestions and comments which gave rise to the opportunity to revise and improve this paper.