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We present a modified model order reduction (MOR) technique for the FFT-based simulation of composite microstructures. It utilizes the earlier introduced MOR technique (Kochmann et al. [2019]), which is based on solving the Lippmann-Schwinger equation in Fourier space by a reduced set of frequencies. Crucial for the accuracy of this MOR technique i...
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... us consider an inhomogeneous periodic microstructure Ω = Ω I ∪ Ω M with inclusions Ω I embedded in a softer matrix material Ω M . One example is a microstructure with one centered spherical inclusion as shown in Figure 1. Considering small strain kinematics, the total strain ε( ¯ x, x) = ¯ ε( ¯ x) + ˜ ε(x) at the macroscopic position ¯ x and the microscopic position x is additively split into the macroscopic part ¯ ε( ¯ x) and the microscopic fluctuating part˜εpart˜ part˜ε(x), respectively. ...
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... a higher number of frequencies, the errors of both solutions converge towards zero. Using the reconstruction algorithm and the compatibility step the solution for the fixed sampling pattern is improved as shown in Figure 10 in the left column. Since the microstructural fields related to the adapted sampling pattern do not have incoherent artifacts, the reconstruction algorithm does not yield any further improvement of the result and is therefore not needed. ...
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... should be mentioned that in this context the earlier given name might be misleading, since the solutions are already compatible. The corresponding solutions are shown in Figure 10 in the centered column. It can be seen, that the micromechanical solution fields which are related to the adapted sampling pattern are still better compared to the solutions based on the fixed sampling pattern after the reconstruction. ...
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... this post-processing step, the macroscopic error ¯ E and microscopic error E are shown in Figure 11. For the fixed sampling pattern, the error refers to the solution after the reconstruction and the compatibility step. ...
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... the adapted sampling pattern, the error corresponds to the compatibility step, only. For both cases, Figure 11 shows the error depending on the percentage of used frequencies R. The difference between the fixed and the adapted sampling pattern is rather small, while the error of the adapted sampling pattern is smaller for R < 6 % but larger for R ≥ 6 %. The range R < 6 % is of much larger interest, since the highest speed-up is gained by a set with a low number of frequencies. ...
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... same investigations could be made using the microstructures with the annular, the elliptical, and the quadratic inclusion shown in Figure 6 and would lead to similar results. Due to that, we just show the sampling patterns and the corresponding microstructural stress fields σ 11 for R = 1.54 % of frequencies considering these microstructures in Figures 12 -14. The results show the relation between the arrangement of the considered frequencies in the adapted sampling pattern and the geometry of the inclusion within the matrix. ...
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... it can be seen, that the results gained by the new approach are always closer to the reference solution than the solutions based on the fixed sampling pattern. In Figure 12, the geometrically adapted sampling pattern of the microstructure with an annular inclusion is shown. This sampling pattern is similar to the adapted sampling corresponding to the circular inclusion; see Figure 8. ...
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... the amount of high frequencies is slightly higher. The adapted sampling pattern corresponding to the elliptical inclusion, shown in Figure 13, differs totally from that. It can be seen, that in direction of the major axis of the ellipse lower frequencies are needed and perpendicular to that higher frequencies are necessary, since a smaller distance needs to be bridged in this direction. ...
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... can be seen, that in direction of the major axis of the ellipse lower frequencies are needed and perpendicular to that higher frequencies are necessary, since a smaller distance needs to be bridged in this direction. As a last example, Figure 14 shows the microstructure with a quadratic inclusion. The sharp edges of this last examined type of inclusion again lead to a totally different set of frequencies. ...
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... an elasto-plastic material behavior of the matrix, the additional material parameters are set to H M = 0.01 GPa as hardening modulus and σ 0 yM = 0.01 GPa as initial yield stress. The investigated microstructure and the prescribed macroscopic strain is presented in Figure 15. For the fixed and the adapted sampling pattern the same amount of frequencies is used. ...
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... the fixed and the adapted sampling pattern the same amount of frequencies is used. Figure 16 shows the microstructural stress fields σ 11 as well as the differences ∆σ 11 to the reference solution for the pure elastic case. As already seen for one inclusion, the error in the solution based on the adapted reduced set of frequencies is significantly lower compared to the solution of the fixed sampling pattern. ...
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... we perform the reconstruction and the compatibility step for the solution based on the fixed sampling pattern and only the compatibility step for the adapted sampling pattern. The corresponding microstructural fields are shown in Figure 17. Here, similar effects as described in Chapter 5.1 for a microstructure with only one inclusion occur: The solution with the adapted sampling pattern is more accurate. ...
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... to that, the geometrically adapted sampling pattern for the microstructure with several elastic inclusions and an elasto-plastic matrix material behavior is the same as for the microstructure with several inclusions and an overall elastic material behvior. Figure 18 shows the microscopic stress field σ 11 corresponding to the fixed and adapted sampling pattern, the reference solution and the absolute difference in the reduced solution compared to the reference solution ∆σ 11 . It can be seen, that the stress difference for the fixed and the adapted sampling pattern is in general higher considering the nonlinear matrix material behavior instead of the purely linear material behavior shown in Figure 16. ...
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... 18 shows the microscopic stress field σ 11 corresponding to the fixed and adapted sampling pattern, the reference solution and the absolute difference in the reduced solution compared to the reference solution ∆σ 11 . It can be seen, that the stress difference for the fixed and the adapted sampling pattern is in general higher considering the nonlinear matrix material behavior instead of the purely linear material behavior shown in Figure 16. Nevertheless, the error in the solution with the adapted sampling pattern is again significantly lower than the error in the solution with the fixed sampling pattern. ...
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... row: Absolute difference in the microstructural stress field ∆σ 11 . Figure 19 shows the macroscopic error ¯ E (left) and the microscopic error E (right) again based on the reduced set of frequencies R for the solution with the fixed and adapted sampling pattern for the elasto-plastic composite. Incorporating R = 1.54 % and considering the fixed sampling pattern, these errors read ¯ E ≈ 34 % and E ≈ 79 %. ...
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... addition, Figure 19 shows that at some point (R ≈ 15 %) the fixed sampling pattern leads to better results than the adapted sampling pattern. This might be related to the elasto-plastic material behavior of the matrix which results in a material behavior which is not that uniform within the matrix as the pure elastic material behavior. ...
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... the reconstruction and compatibility step for the solution of the fixed sampling pattern and only the compatibility step for the solution of the adapted sampling pattern leads to the results given in Figures 21 and 22. Figure 21 shows the microstructural stress field σ 11 and Figure 22 shows the accumulated plastic strain field ε acc p , respectively. ...
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... the reconstruction and compatibility step for the solution of the fixed sampling pattern and only the compatibility step for the solution of the adapted sampling pattern leads to the results given in Figures 21 and 22. Figure 21 shows the microstructural stress field σ 11 and Figure 22 shows the accumulated plastic strain field ε acc p , respectively. Considering the fixed sampling pattern, it can be seen, that the calculated stress within the inclusions is improved by the reconstruction and the compatibility step, while the stress within the elasto-plastic matrix is not improved significantly. ...
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... is related to the accumulated plastic strain field, shown in Figure 22, which is also not improved by these post-processing steps. As shown in Figure 21, the microstructural stress field related to the solution with the adapted sampling pattern is slightly improved by solving the Lippmann-Schwinger equation once with the full set of frequencies. Middle row: Corresponding microstructural stress field σ 11 incorporating the reconstruction and compatibility step for the solution of the fixed sampling pattern and only the compatibility step for the solution of the adapted sampling pattern and reference stress field computed with the full set of frequencies. ...
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... us consider an inhomogeneous periodic microstructure Ω = Ω I ∪ Ω M with inclusions Ω I embedded in a softer matrix material Ω M . One example is a microstructure with one centered spherical inclusion as shown in Figure 1. Considering small strain kinematics, the total strain ε( ¯ x, x) = ¯ ε( ¯ x) + ˜ ε(x) at the macroscopic position ¯ x and the microscopic position x is additively split into the macroscopic part ¯ ε( ¯ x) and the microscopic fluctuating part˜εpart˜ part˜ε(x), respectively. ...
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... a higher number of frequencies, the errors of both solutions converge towards zero. Using the reconstruction algorithm and the compatibility step the solution for the fixed sampling pattern is improved as shown in Figure 10 in the left column. Since the microstructural fields related to the adapted sampling pattern do not have incoherent artifacts, the reconstruction algorithm does not yield any further improvement of the result and is therefore not needed. ...
Context 22
... should be mentioned that in this context the earlier given name might be misleading, since the solutions are already compatible. The corresponding solutions are shown in Figure 10 in the centered column. It can be seen, that the micromechanical solution fields which are related to the adapted sampling pattern are still better compared to the solutions based on the fixed sampling pattern after the reconstruction. ...
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... this post-processing step, the macroscopic error ¯ E and microscopic error E are shown in Figure 11. For the fixed sampling pattern, the error refers to the solution after the reconstruction and the compatibility step. ...
Context 24
... the adapted sampling pattern, the error corresponds to the compatibility step, only. For both cases, Figure 11 shows the error depending on the percentage of used frequencies R. The difference between the fixed and the adapted sampling pattern is rather small, while the error of the adapted sampling pattern is smaller for R < 6 % but larger for R ≥ 6 %. The range R < 6 % is of much larger interest, since the highest speed-up is gained by a set with a low number of frequencies. ...
Context 25
... same investigations could be made using the microstructures with the annular, the elliptical, and the quadratic inclusion shown in Figure 6 and would lead to similar results. Due to that, we just show the sampling patterns and the corresponding microstructural stress fields σ 11 for R = 1.54 % of frequencies considering these microstructures in Figures 12 -14. The results show the relation between the arrangement of the considered frequencies in the adapted sampling pattern and the geometry of the inclusion within the matrix. ...
Context 26
... it can be seen, that the results gained by the new approach are always closer to the reference solution than the solutions based on the fixed sampling pattern. In Figure 12, the geometrically adapted sampling pattern of the microstructure with an annular inclusion is shown. This sampling pattern is similar to the adapted sampling corresponding to the circular inclusion; see Figure 8. ...
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... the amount of high frequencies is slightly higher. The adapted sampling pattern corresponding to the elliptical inclusion, shown in Figure 13, differs totally from that. It can be seen, that in direction of the major axis of the ellipse lower frequencies are needed and perpendicular to that higher frequencies are necessary, since a smaller distance needs to be bridged in this direction. ...
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... can be seen, that in direction of the major axis of the ellipse lower frequencies are needed and perpendicular to that higher frequencies are necessary, since a smaller distance needs to be bridged in this direction. As a last example, Figure 14 shows the microstructure with a quadratic inclusion. The sharp edges of this last examined type of inclusion again lead to a totally different set of frequencies. ...
Context 29
... an elasto-plastic material behavior of the matrix, the additional material parameters are set to H M = 0.01 GPa as hardening modulus and σ 0 yM = 0.01 GPa as initial yield stress. The investigated microstructure and the prescribed macroscopic strain is presented in Figure 15. For the fixed and the adapted sampling pattern the same amount of frequencies is used. ...
Context 30
... the fixed and the adapted sampling pattern the same amount of frequencies is used. Figure 16 shows the microstructural stress fields σ 11 as well as the differences ∆σ 11 to the reference solution for the pure elastic case. As already seen for one inclusion, the error in the solution based on the adapted reduced set of frequencies is significantly lower compared to the solution of the fixed sampling pattern. ...
Context 31
... we perform the reconstruction and the compatibility step for the solution based on the fixed sampling pattern and only the compatibility step for the adapted sampling pattern. The corresponding microstructural fields are shown in Figure 17. Here, similar effects as described in Chapter 5.1 for a microstructure with only one inclusion occur: The solution with the adapted sampling pattern is more accurate. ...
Context 32
... to that, the geometrically adapted sampling pattern for the microstructure with several elastic inclusions and an elasto-plastic matrix material behavior is the same as for the microstructure with several inclusions and an overall elastic material behvior. Figure 18 shows the microscopic stress field σ 11 corresponding to the fixed and adapted sampling pattern, the reference solution and the absolute difference in the reduced solution compared to the reference solution ∆σ 11 . It can be seen, that the stress difference for the fixed and the adapted sampling pattern is in general higher considering the nonlinear matrix material behavior instead of the purely linear material behavior shown in Figure 16. ...
Context 33
... 18 shows the microscopic stress field σ 11 corresponding to the fixed and adapted sampling pattern, the reference solution and the absolute difference in the reduced solution compared to the reference solution ∆σ 11 . It can be seen, that the stress difference for the fixed and the adapted sampling pattern is in general higher considering the nonlinear matrix material behavior instead of the purely linear material behavior shown in Figure 16. Nevertheless, the error in the solution with the adapted sampling pattern is again significantly lower than the error in the solution with the fixed sampling pattern. ...
Context 34
... row: Absolute difference in the microstructural stress field ∆σ 11 . Figure 19 shows the macroscopic error ¯ E (left) and the microscopic error E (right) again based on the reduced set of frequencies R for the solution with the fixed and adapted sampling pattern for the elasto-plastic composite. Incorporating R = 1.54 % and considering the fixed sampling pattern, these errors read ¯ E ≈ 34 % and E ≈ 79 %. ...
Context 35
... addition, Figure 19 shows that at some point (R ≈ 15 %) the fixed sampling pattern leads to better results than the adapted sampling pattern. This might be related to the elasto-plastic material behavior of the matrix which results in a material behavior which is not that uniform within the matrix as the pure elastic material behavior. ...
Context 36
... the reconstruction and compatibility step for the solution of the fixed sampling pattern and only the compatibility step for the solution of the adapted sampling pattern leads to the results given in Figures 21 and 22. Figure 21 shows the microstructural stress field σ 11 and Figure 22 shows the accumulated plastic strain field ε acc p , respectively. ...
Context 37
... the reconstruction and compatibility step for the solution of the fixed sampling pattern and only the compatibility step for the solution of the adapted sampling pattern leads to the results given in Figures 21 and 22. Figure 21 shows the microstructural stress field σ 11 and Figure 22 shows the accumulated plastic strain field ε acc p , respectively. Considering the fixed sampling pattern, it can be seen, that the calculated stress within the inclusions is improved by the reconstruction and the compatibility step, while the stress within the elasto-plastic matrix is not improved significantly. ...
Context 38
... is related to the accumulated plastic strain field, shown in Figure 22, which is also not improved by these post-processing steps. As shown in Figure 21, the microstructural stress field related to the solution with the adapted sampling pattern is slightly improved by solving the Lippmann-Schwinger equation once with the full set of frequencies. Middle row: Corresponding microstructural stress field σ 11 incorporating the reconstruction and compatibility step for the solution of the fixed sampling pattern and only the compatibility step for the solution of the adapted sampling pattern and reference stress field computed with the full set of frequencies. ...
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