Coupled SelfSim and genetic programming for non-linear material constitutive modelling

Figures & data

Figure 1. Simplified flowchart of SelfSim framework.

Figure 2. Comparison of the GP programme structures. (a) LGP (b) Tree-based GP [32].

Figure 3. An excerpt of a linear genetic programme.

Figure 4. Simplified flowchart of the LGP.

Table 1. Parameter settings for the LGP algorithm.

Parameter	Settings
Function set	+, -, ×, /, √
Population size	500, 1000
Maximum number of tournaments	10,000
Maximum programme size	128
Initial programme size	80
Crossover rate (%)	50, 95
Homologous crossover (%)	95
Mutation rate (%)	50, 95
Block mutation rate (%)	30
Instruction mutation rate (%)	30
Data mutation rate (%)	40
Number of demes	20

Figure 5. Experimental vs. predicted stress–strain relationship for soils under unconfined compression.

Figure 6. Experimental vs. predicted stress–strain relationship for aorta.

Figure 7. Simplified pseudo-code of SelfSim–GP algorithm.

Figure 8. Space truss structure used in case study.

Figure 9. Global force–displacement response of the truss structure.

Figure 10. Stress–strain behaviour modelling using ANN.

Figure 11. A comparison of reference model, ANN-based model and GP model.

Figure 12. Stress–strain distribution of the whole structure. (a) Reference material model, (b) GP-based material model derived by SelfSim–GP methodology.

Table A1. The optimum LGP programmes of the case studies which can be run in the Discipulus interactive evaluator mode or can be compiled in C++ environment. (Note: v[0], represents ε).

Case I (soil compression model)	Case II (structural aortic model)	Case III (ANN-based material model)
float Discipulus C Function (float v[])	float Discipulus C Function (float v[])	float Discipulus C Function (float v[])
{	{	{
double f[8];	double f[8];	double f[8];
double tmp = 0;	double tmp = 0;	double tmp = 0;
f[1]=f[2]=f[3]=f[4]=f[5]= f[6]=f[7]=f[8]= 0;	f[1]=f[2]=f[3]=f[4]=f[5]=f[6]=f[7]=f[8]= 0;	f[1]=f[2]=f[3]=f[4]=f[5]=f[6]=f[7]=f[8]= 0;
f[0]=v[0];	f[0]=v[0];	f[0]=v[0];
l0: f[0]/=0.25;	l0: f[0]+=f[0];	l0: f[0]*=8;
l1: f[0]−=0.5;	l1: f[0]+=f[0];	l1: f[0]=sqrt(f[0]);
l2: f[0]*=f[0];	f[1]+=f[0];	f[1]−=f[0];
l3: f[0]+=f[0];	l2: f[0]−=0.5;	l2: f[0]−=0.25;
f[0]+=f[0];	l3: l4: f[0]/=9;	l3: f[0]*=f[0];
l4: f[0]+=f[0];	l5: f[0]+=v[0];	f[1]+=f[0];
l5: f[0]+=0.5;	l6: f[0]*=f[0];	l4: f[1]+=f[0];
l6: f[0]−=4;	f[0]=sqrt(f[0]);	f[0]+=f[1];
l7: f[0]*=v[0];	l7: f[0]*=f[1];	l5: f[0]=fabs(f[0]);
l8:	f[0]=sqrt(f[0]);	f[0]−=f[1];
l9:	l8:	l6: f[1]−=f[0];
return f[0];	l9:	f[0]*=f[0];
}	return f[0];	l7: f[0]*=f[0];
	}	f[0]+=f[1];
		l8:
		l9:
		return f[0];
		}

Gandomi AH, Alavi AH, Hosseini SS. A discussion on ‘Genetic programming for retrieving missing information in wave records along the west coast of India’. Appl. Ocean Res. 2008;30:338–339. 2007;29:99–111.

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