Fractional Temperature-Dependent BEM for Laser Ultrasonic Thermoelastic Propagation Problems of Smart Nanomaterials
Abstract
:1. Introduction
2. Formulation of the Problem
3. Boundary Conditions
4. Boundary Element Implementation
5. Numerical Results and Discussion
6. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
ᾱ | Coefficient of thermal expansion | Fα | Body force vector |
a | Fractional-order parameter | F* | Point force kernel function |
δαβ | Kronecker delta function | f | Piezoelectric coefficient |
λ & µ | Lamé elastic constants | J(τ) | Non-Gaussian temporal profile |
ρE | Volume electric charge density | J0 | Total energy intensity |
η | Couple-stress parameter | k | Thermal conductivity |
σαβ | Total force-stress tensor | kα | Mean curvature vector |
σ(αβ) | Symmetric force-stress tensor | kαβ | Pseudo mean curvature tensor |
σ[αβ] | Skew-symmetric force-stress tensor | l | The material length scale parameter |
τ | Time | Mi | True couple-stress vector |
τ1 | Laser pulse time characteristic | Mkj | Pseudo couple-stress tensor |
φ | Electric potential | m | Couple-traction |
Ω | Rotation | nα | Outward unit normal vector |
A | Non-symmetric dense matrix | Pα | Polarization of piezoelectric material |
B | Known boundary values vector | Q | External heat source |
C* | Point couple kernel function | Q* | Point heat source kernel function |
Dα | Electric displacement | q | Normal flux |
d | Normal electric displacement | qα | Heat flux vector |
E | Young’s modulus | R | Irradiated surface absorptivity |
Eα | Electric field | R* | Point electrical source kernel function |
eαβ | 2D permutation symbol | T | Temperature |
eijk | 3D Levi-Civita permutation symbol | tI | Generalized tractions |
e | Electric permittivity | tα | Force-traction vector |
er | Relative permittivity | uα | Displacement vector |
e0 | Vacuum permittivity | v | Poisson ratio |
X | Unknown boundary values vector |
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T(°C) | 0 | 500 | 900 |
C(J/kg) °K | 385 | 433 | 480 |
ρ(kg/m3) | 8930 | 8686 | 8458 |
l | Method | Iter. | CPU Time | Rr | Err. |
---|---|---|---|---|---|
0.01 | SCAS-GMRES | 30 | 0.0119 | 1.96 × 10−7 | 1.48 × 10−9 |
FMDTS | 60 | 0.0564 | 5.50 × 10−7 | 1.72 × 10−7 | |
UC-RSCSCS | 70 | 0.0730 | 7.02 × 10−7 | 2.50 × 10−6 | |
0.1 | SCAS-GMRES | 40 | 0.0538 | 0.19 × 10−6 | 2.06 × 10−8 |
FMDTS | 90 | 0.2239 | 1.72 × 10−5 | 4.52 × 10−6 | |
UC-RSCSCS | 120 | 0.3764 | 1.16 × 10−4 | 0.58 × 10−5 | |
1.0 | SCAS-GMRES | 60 | 0.1758 | 2.22 × 10−5 | 1.80 × 10−7 |
FMDTS | 270 | 0.7940 | 1.80 × 10−4 | 3.62 × 10−5 | |
UC-RSCSCS | 280 | 0.8950 | 1.22 × 10−3 | 4.60 × 10−4 |
l | BEM | FEM | Analytical | |||
---|---|---|---|---|---|---|
0.01 | −0.04766 × 10−12 | −0.01847 × 10−12 | −0.04769 × 10−12 | −0.01850 × 10−12 | −0.04767 × 10−12 | −0.01848 × 10−12 |
0.1 | −0.02452 × 10−12 | −0.02113 × 10−12 | −0.02455 × 10−12 | −0.02116 × 10−12 | −0.02453 × 10−12 | −0.02114 × 10−12 |
1.0 | −0.01984 × 10−12 | −0.02582 × 10−12 | −0.01987 × 10−12 | −0.02586 × 10−12 | −0.01985 × 10−12 | −0.02583 × 10−12 |
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Fahmy, M.A. Fractional Temperature-Dependent BEM for Laser Ultrasonic Thermoelastic Propagation Problems of Smart Nanomaterials. Fractal Fract. 2023, 7, 536. https://doi.org/10.3390/fractalfract7070536
Fahmy MA. Fractional Temperature-Dependent BEM for Laser Ultrasonic Thermoelastic Propagation Problems of Smart Nanomaterials. Fractal and Fractional. 2023; 7(7):536. https://doi.org/10.3390/fractalfract7070536
Chicago/Turabian StyleFahmy, Mohamed Abdelsabour. 2023. "Fractional Temperature-Dependent BEM for Laser Ultrasonic Thermoelastic Propagation Problems of Smart Nanomaterials" Fractal and Fractional 7, no. 7: 536. https://doi.org/10.3390/fractalfract7070536