Metamodel-Based Inverse Design of a Composite Material with Prescribed Interval Effective Elastic Properties

BELUCH, Witold; PTASZNY, Jacek; HATŁAS, Marcin

doi:10.2478/ama-2025-0033

Figures & Tables

Granular Computational Inverse Design scheme

The RVE model of the fibre-reinforced composite material: a) unit cell geometry, b) FEM mesh

Results of the multi-objective optimisation

Run	I_H	Pareto point	p¯1=Em[GPa]	p¯2=vm[−]	p¯3=Ef[GPa]	p¯4=vf[−]	p¯5=f[−]	f1(p¯)[GPa]	f2(p¯)[−]	C¯22(p¯)[GPa]	δ_w, δ_c [%]
1	-0.26058	min f1(p¯)	[6.21597, 6.45371]	[0.32362, 0.32760]	[149.183, 180.177]	[0.21958, 0.30889]	[0.32641, 0.34275]	2.9865E-03	3.9623E-02	[15.197072, 16.799410]	1.0994E-021.4612E-01
1	-0.26058	max f2(p¯)	[5.65513, 6.52451]	[0.30761, 0.32362]	[377.437, 450.000]	[0.31219, 0.33490]	[0.30399, 0.36323]	2.4308E+00	1.4490E-01	[13.106417, 18.035232]	2.6823E+002.0805E+02
2	-0.28839	min f1(p¯)	[6.64138, 6.87106]	[0.36113, 0.36698]	[71.940, 162.432]	[0.26203, 0.31893]	[0.22596, 0.23219]	2.9716E-03	3.1145E-02	[15.201644, 16.802470]	1.2856E-025.1625E-02
2	-0.28839	max f2(p¯)	[5.78269, 6.87106]	[0.32687, 0.34218]	[334.408, 450.000]	[0.29318, 0.35000]	[0.25825, 0.28928]	2.2397E+00	1.5310E-01	[13.032410, 17.363930]	5.0114E+001.7072E+02
3	-0.29693	min f1(p¯)	[6.02157, 6.24235]	[0.30924, 0.32460]	[313.892, 323.541]	[0.20476, 0.23224]	[0.34545, 0.35226]	9.1474E-04	2.4120E-02	[15.199641, 16.799159]	3.7500E-033.0125E-02
3	-0.29693	max f2(p¯)	[5.94323, 6.93317]	[0.30700, 0.32460]	[357.547, 450.000]	[0.28208, 0.35000]	[0.27794, 0.32512]	2.3664E+00	1.6499E-01	[13.03876217.763848]	3.7418E+001.9532E+02
4	-0.28615	min f1(p¯)	[5.67805, 5.81318]	[0.30305, 0.32223]	[318.653, 450.000]	[0.21592, 0.30478]	[0.38134, 0.38597]	4.6047E-03	2.2522E-02	[15.201917, 16.795817]	7.0812E-033.8125E-01
4	-0.28615	max f2(p¯)	[5.26435, 6.37166]	[0.34681, 0.36387]	[345.252, 450.000]	[0.31243, 0.35000]	[0.28161, 0.31635]	2.4268E+00	1.7065E-01	[13.389715, 18.416232]	6.0642E-012.1416E+02
5	-0.27313	min f1(p¯)	[5.49730, 5.83237]	[0.33979, 0.34532]	[377.152, 450.000]	[0.20000, 0.24683]	[0.33794, 0.34838]	6.2008E-04	5.2240E-02	[15.200134, 16.799395]	1.4719E-034.6188E-02
5	-0.27313	max f2(p¯)	[5.20406, 6.14267]	[0.31508, 0.33451]	[359.835, 450.000]	[0.30955, 0.33613]	[0.33212, 0.37497]	2.4372E+00	1.5643E-01	[13.098494, 18.034225]	2.7103E+002.0848E+02
6	-0.30318	min f1(p¯)	[6.74116, 7.13123]	[0.34609, 0.35073]	[326.580, 388.616]	[0.22717, 0.31078]	[0.22181, 0.23521]	2.8838E-03	4.6350E-02	[15.201307, 16.802570]	1.2116E-027.8937E-02
6	-0.30318	max f2(p¯)	[6.84060, 8.00000]	[0.31912, 0.34471]	[313.585, 450.000]	[0.28905, 0.35000]	[0.20547, 0.24324]	2.4610E+00	1.8884E-01	[13.588805, 18.660209]	7.7817E-012.1696E+02
7	-0.29861	min f1(p¯)	[6.07599, 6.13976]	[0.35545, 0.36251]	[66.934, 103.602]	[0.23487, 0.27334]	[0.29319, 0.31667]	1.0141E-02	1.0628E-02	[15.204597, 16.790961]	1.3881E-028.5225E-01
7	-0.29861	max f2(p¯)	[6.16085, 7.23273]	[0.30000, 0.31959]	[311.748, 381.206]	[0.24644, 0.27261]	[0.28919, 0.33169]	2.3757E+00	1.7365E-01	[13.514604, 18.474349]	3.4522E-022.0998E+02
8	-0.28615	min f1(p¯)	[7.06127, 7.38220]	[0.30000, 0.30279]	[357.869, 392.536]	[0.28429, 0.32087]	[0.27701, 0.29914]	3.8161E-03	-2.796E-02	[15.201885, 16.803318]	1.6259E-028.9562E-02
8	-0.28615	max f2(p¯)	[5.11271, 6.11818]	[0.37069, 0.38602]	[326.957, 450.000]	[0.32087, 0.35000]	[0.24097, 0.27701]	2.1827E+00	1.5331E-01	[13.333662, 17.931799]	2.2954E+001.8738E+02
9	-0.29202	min f1(p¯)	[6.98408, 7.14029]	[0.32483, 0.33341]	[89.1451, 110.587]	[0.27858, 0.34763]	[0.27141, 0.28943]	2.6195E-03	2.6035E-02	[15.202586, 16.799585]	6.7844E-031.8756E-01
9	-0.29202	max f2(p¯)	[5.12698, 6.16768]	[0.36979, 0.38819]	[293.851450.000]	[0.30270, 0.33132]	[0.25029, 0.28474]	2.3811E+00	1.7227E-01	[13.529283, 18.496622]	8.0953E-022.1046E+02
10	-0.29388	min f1(p¯)	[6.41585, 6.79624]	[0.37605, 0.37959]	[50.000, 71.4883]	[0.20603, 0.24717]	[0.21665, 0.22781]	2.9710E-03	3.5399E-02	[15.199480, 16.802953]	7.6031E-032.1706E-01
10	-0.29388	max f2(p¯)	[6.41585, 7.39224]	[0.34290, 0.36104]	[346.451, 450.000]	[0.30325, 0.34305]	[0.22490, 0.25735]	2.1655E+00	1.6221E-01	[14.412384, 18.817226]	3.8425E+001.7530E+02

Exemplary results of the single-objective optimisation for φ1 = 0_75, φ2=0_25

Rank	p¯1=Em[GPa]	p¯2=vm[−]	p¯3=Ef[GPa]	p¯4=vf[−]	p¯5=f[−]	g1(p¯)[GPa]	g2(p¯)[−]	fg(p¯)	C¯22(p¯)[GPa]	δ_v, δ_c [%]
1	[5.29892, 5.54251]	[0.36931, 0.37214]	[303.429, 370.799]	[0.26779, 0.34706]	[0.29828, 0.31968]	7.7079E-04	2.8280E-02	-6.4919E-03	[15.200105, 16.800763]	2.7100E-034.1120E-02
2	[6.58579, 6.84576]	[0.31106, 0.31661]	[179.078, 257.930]	[0.23188, 0.34675]	[0.31161, 0.32033]	1.0045E-02	4.3329E-02	-3.2986E-03	[15.195634, 16.790953]	4.1916E-022.9256E-01
3	[6.02565, 6.17328]	[0.32825, 0.33579]	[260.926, 321.946]	[0.26363, 0.25501]	[0.33863, 0.31755]	4.2133E-03	2.4606E-02	-2.9916E-03	[15.202722, 16.803215]	1.8553E-023.0813E-02
4	[5.75597, 6.09119]	[0.37103, 0.36848]	[80.847, 99.605]	[0.22293, 0.29563]	[0.29018’0.30173]	4.9521E-03	2.5470E-02	-2.6534E-03	[15.204895, 16.800749]	1.7638E-022.5913E-01
5	[5.79374, 6.38056]	[0.37613, 0.37806]	[317.142, 421.272]	[0.33603, 0.34437]	[0.23377, 0.23645]	2.3205E-03	1.3425E-02	-1.6159E-03	[15.199783, 16.802310]	6.5406E-031.5794E-01
30	[7.34717, 7.41631]	[0.34548, 0.34419]	[98.303, 450,000]	[0.28926, 0.29484]	[0.20000, 0.22936]	5.8223E-02	1.1523E-02	4.0787E-02	[15.194355, 16.857949]	1.6345E-013.9746E+00

Polynomial coefficients for C22RS and quality metrics

β₀	2.663E+01
β_k	β₁ = -8.969E-01; β₂ = -1.410E+02; β₃ = -3.450E-04; β₄ = -3.622E-01; β₅ = -4.214E+01
β_kl	β₁₁ = 6.144E+00; β₁₂ = 5.427E-04; β₁₃ = 1.586E-01; β₁₄ = 4.170E+00; β₁₅ = 6.277E-03; β₂₂ = 2.490E+00; β₂₃ = 5.768E+01; β₂₄ = 4.311E-03; β₂₅ = 1.405E-02; β₃₃ = 3.669E+00; β₃₄ = -1.067E-02; β₃₅ = 2.056E+02; β₄₄ = -1.086E-05; β₄₅ = -8.753E-01; β₅₅ = 3.672E+01
R²	0.99586
PRESS	23.23325
σ_est	0.35107

The best results of the single-objective optimisation for different weighting coefficients

Variant	A	B	C
Weighting coefficients	φ₁ = 0.25, φ2 = 0.75	φ₁ = 0.5, φ₂ = 0.5	φ₁ = 0.75, φ₂ = 0.25
p¯1=Em[GPa]	[6.05064, 6.46002]	[4.91259, 5.15732]	[5.29892, 5.54251]
p¯2=vm[−]	[0.37837, 0.38391]	[0.34833, 0.35504]	[0.36931, 0.37214]
p¯3=Ef[GPa]	[443.933, 354.949]	[298.173, 331.484]	[303.429, 370.799]
p¯4=vf[−]	[0.30482, 0.31628]	[0.24757, 0.27883]	[0.26779, 0.34706]
p¯5=f[−]	[0.20564, 0.21752]	[0.37770, 0.38936]	[0.29828, 0.31968]
g1(p¯)[GPa]	1.0987E-02	1.6975E-03	7.7079E-04
g2(p¯)[−]	5.5390E-02	4.0799E-02	2.8280E-02
fg(p¯)	-3.8796E-02	-1.9551E-02	-6.4919E-03
avg[ fg(p¯) ]	-1.6084E-02	-5.5497E-03	1.6779E-02
σ[ fg(p¯) ]	9.9484E-03	1.1636E-02	5.7618E-02
C¯22(p¯)[GPa]	[15.210668, 16.797376]	[15.200626, 16.801578]	[15.200105, 16.800763]
δ_w [%]	2.5140E-02	6.8900E-03	2.7100E-03
δ_c [%]	8.3075E-01	5.9500E-02	4.1120E-02

Metamodel-Based Inverse Design of a Composite Material with Prescribed Interval Effective Elastic Properties

Figures & Tables

Fig. 1.

Fig. 2.

Fig. 3.

Results of the multi-objective optimisation

Exemplary results of the single-objective optimisation for φ1 = 0_75, φ2=0_25

Polynomial coefficients for C22RS and quality metrics

The best results of the single-objective optimisation for different weighting coefficients

Paradigm

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