Profiles resulting from TE-FEM (red open circle) and TE-FFT (blue open circle) for 32×32×32 resolution on stress profile calculated from Eshelby (black solid line). Stress field (von Mises equivalent stress) on the cross-section of the microstructure. a) prediction from TE-FEM for random orientation set 1 to 5, respectively, and (b) displays relative results from TE-FFT. a) Calculated differences in stress between two methods. Probability density functions for stress resulting from TE-FEM (red) and TE-FFT (blue) Figure 13.
Probability plots of normalized stress due to TE-FEM (red) and TE-FFT (blue) Figure 14. Threshold selection plots and average residual life plots for the stress resulting from (a) TE-FEM. Stress fields on the cross-section of the microstructure. a) prediction of TE-FEM for arbitrary orientation set 1 to 5 respectively, and (b) shows relative results of TE-FFT.
Introduction
In this regard, the differences between FEM and FFT in predicting the elastic response under thermal loading with respect to the upper tail were investigated, with reference to extreme value theory, to verify the similarities and peculiarities of the two methods. First, the stress distributions derived from the FE and FFT methods are validated by comparison with the analytical solution of the Eshelby problem for a spherical inclusion embedded in an infinite matrix [8]. For an isotropic case with a cubic crystal structure, the differences are analyzed according to the theory of extreme values.
By doing so, similarities and distinctiveness in elastic response between FEM and FFT are quantified.
Theoretical Background
- Thermoelasticity
- Thermoelastic FEM
- Thermoelastic FFT
- Extreme Value Analysis
- Introduction to Extreme Value Analysis
Hotspot stress or elastic strain energy density (EED) in the upper tail of the distribution are the main areas of our interest, due to their important role in creep and fatigue characteristics under temperature variation. The elongation ∆𝑙 can be expressed as ∆𝑙 = 𝐹𝑙F/𝐸𝐴F, where E is the elastic modulus (also known as stiffness constant, 𝐶). By considering these equations, a basic formula for elastic deformation can be expressed as Hooke's law [22].
The FEM approach is recognized as one of the powerful numerical methods that can be applied to numerous engineering analyzes [25, 26]. The domain integral of the weak-field equation is then decomposed into integrals over each element, and the solution is approximated by means of a simple polynomial form in the element. The fundamental framework of FFT was hypothesized by Moulinec and Suquet [15-17] which gave us guidance for introducing the solution of the local problem of an inhomogeneous elastic medium.
Since the strain field is related to the displacement field 𝑢M(𝐱) as:. Since the polarization field in real space 𝜏)*(𝐱) is unknown, 𝜀)*(𝐱) is guessed using anti-transformation and symmetrization of Eq.(25), and then applied to Eq.(20) for a new guess at the polarization field. Conventionally, the prediction of the mechanical response using numerical simulation usually relied on the incomplete descriptions, such as histogram, or focused on the mean and standard deviation of data, ignoring the extreme values in the upper/lower tail of the distribution. Although such descriptions provide the fundamental information about the stress and strain range from simulations, effort must be made to study substantial statistical information to understand the weighty values of the upper tail, for example hotspot stress.
Two commonly used methodologies in extreme value analysis. a) the block maxima and (b) the Peaks-over-threshold (POT). The associated three parameters of the GPD play an important role, where the shape parameter (𝜉) indicates the spread, the scale parameter (𝜎) implies the length and the location parameter (𝑢) indicates the threshold of the tail distribution. It is noteworthy that 𝜎 and 𝜉 are functions of the threshold (𝜇), so the properties of these estimators depend significantly on the behavior of the distribution tail.
The threshold selection graph gave us guidelines for the selection of excesses in terms of GPD stability properties.
Eshelby’s Inclusion Problem
Introduction
From eq.(34), 𝐸[𝑋 − 𝜇|𝑋 > 𝜇] is a linear function of 𝜇, which yields a graphical identification for appropriate threshold in the modeling of extreme values via the GPD.
Simulation parameters
Here a volume fraction of spherical inclusion is maintained at 0.8%, so the radius of the voxelized sphere is roughly determined at 4 and 8 voxel elements for dimensions 32 x 32 x 32 and 64 x 64 x 64. The structural elements of a linear 3D -brick element with eight nodes and one Gaussian point (C3D8R) are used for the FEM-based simulation, corresponding to a periodic unit cell of the one-point FFT-based simulations.
Results & Discussion
- Stress distribution
- Stress profiles
Stress distribution on the cross-section (the section at the center of z-axis with x-axis horizontal, y-axis vertical) predicted from TE-FEM and TE-FFT is shown in Fig. 3 and 4, where fig. 3 shows the results corresponding to the microstructure with the size 32×32×32 and fig.4 shows its higher resolution, with the size 64×64×64. In fig. 3 shows each diagonal stress predicted from (a), (d) TE-FEM large oscillation-like mosaic pattern, especially near inclusion-matrix interfaces, while (b), (d) TE-FFT represents monotonically changing distribution. Stress profiles predicted from TE-FEM and TE-FFT for 32×32×32 resolution are superimposed on respectively calculated Eshelby stress profile (black solid line).
Here, stress value resulting from TE-FEM is represented as red open square, and from TE-FFT is depicted as blue open circle. According to the Eshelby's solution, theoretical homogeneous stress inside the inclusion is - 376.05 MPa and corresponding average 𝜎77 predicted from TE-FEM and TE-FFT is -370.99 and - 375.93 MPa, respectively. This implies that prediction of TE-FFT is much closer to the analytical solution of Eshelby.
Cross-sectional stress fields (the section in the middle of the z-axis with the x-axis. horizontal, the y-axis vertical) predicted by TE-FEM and TE-FFT for a 32×32×32 resolution. In fig. 5 (a), the 𝜎77 profile calculated from the TE-FFT shows a relatively monotonic distribution as observed in the previous section, while a certain amount of fluctuation is observed in the TE-FEM prediction. It is observed that variations performed in 32×32×32 resolution became weaker with increasing resolution, which implies that 64×64×64 resolution is very reasonable to approximate Eshelby's solution.
In terms of computational time, the FFT-based simulation for a homogeneous Eshelby case of size 32 × 32 × 32 and 64 × 64 × 64 in an apple iMac (4-core Intel Core i5 with 3.2 GHz) took only less than a minute. Profiles derived from TE-FEM (red open circle) and TE-FFT (blue open circle) for 64 × 64 × 64 resolution on the stress profile calculated from Eshelby (black solid line).
Thermoelastic Response of a Polycrystalline Material
- Introduction
- Model description
- Isotropic Case
- Material Selection
- Comparison of local response: von Mises equivalent stress distribution
- Statistical Extreme value Techniques
- Anisotropic Case
- Material Selection
- von Mises equivalent stress distribution
- Application of Eshelby problem to anisotropic case
Figure 8 visualizes the stress distribution on the external surfaces of the microstructure, where (a) represents the results of the TE-FEM approach for the orientation set 1 to 5, and (b) shows the corresponding results of the TE-FFT approach. In both graphs, TE-FEM and TE-FFT results were marked with red and blue symbols, respectively. In the following, the stress probability density functions derived from TE-FEM and TE-FFT for 1 to 5 sets of test microstructure are overlaid as shown in Figure 12.
The histogram of stress datasets calculated from TE-FEM and TE-FFT is shown as red and blue, respectively. It is noteworthy that the mean, maximum and standard deviation of stress due to TE-FEM is higher than TE-FFT for every five cases, indicating that there is a potential distinction between two methods involved in extreme values of stress distribution. Probability density functions for stress due to TE-FEM (red) and TE-FFT (blue). a) Scatterplot and (b) boxplot of strain versus normalized grain size.
The probability plots of the comprehensive stress data sets resulting from both methods are plotted on the same graph as shown in Fig.13, where the open red circle represents the stress from the TE-FEM prediction and the open blue circle indicates the stress from TE-FFT prediction. Fig.14 shows the threshold selection plots and mean residual lifetime plots for the datasets from (a) TE-FEM prediction and (b) TE-FFT prediction, respectively. Since smaller shape parameter indicates a shorter tail, (b) TE-FFT has much shorter tail than TE-FEM.
Threshold selection plots and mean residual lifetime plots for stress derived from (a) TE-FEM and TE-FFT. The probability density function for TE-FEM and TE-FFT calculations with set 1 to 5 is presented in Figure 19. In addition, as shown in the descriptive statistics summarized in Table 4, the mean, maximum and standard deviation of the stress that derived from TE-FFT greater than TE-FEM.
In (a)~(d), on the left, two figures represent cross-sectional stress fields derived from TE-FEM and TE-FFT.
Conclusion
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