Mixed Static and Dynamic Optimization of Four-Parameter Functionally Graded Completely Doubly Curved and Degenerate Shells and Panels Using GDQ Method

Table 2

Optimization of the six different structures of Figure 4: (a) square plate, (b) cylindrical panel, (c) conical shell, (d) toroidal shell panel, (e) catenoidal panel, and (f) elliptic paraboloid. 𝜂=1: optimization of the first frequency 𝑓₁; 𝜂=0: optimization of maximum bending displacement 𝑤_max=𝑢_3max; 𝜂=0.5: optimization of the first frequency 𝑓₁ and maximum bending displacement 𝑤_max=𝑢_3max.

Method	𝑎	𝑏	𝑐	𝑝	Final frequency	Final displacement	Final fitness	Computational time (min)
Rectangular plate
𝜂=1: target frequency 𝑓1𝑇=268.753 Hz
MC	0.05582	−0.56831	17.86455	9.14279	265.36499	1.8270𝐸−05	1.2606𝐸−02	25
GA	0.08800	−0.06160	5.02440	9.53080	268.81630	1.7493𝐸−05	2.3583𝐸−04	42
PSO	−0.25829	−0.13011	0.67170	3.43176	268.71000	1.7466𝐸−05	1.5970𝐸−04	31
Cylindrical panel
𝜂=0: target displacement 𝑤max𝑇=𝑢3max𝑇=1.1646⋅10^−5m
MC	1.11895	1.10588	1.09493	13.33798	186.78559	1.1596𝐸−05	4.3053𝐸−03	32
GA	0.12710	−0.33040	6.11930	3.03030	202.41150	1.1668𝐸−05	1.8770𝐸−03	14
PSO	0.35190	−0.26277	6.90518	1.75727	197.75230	1.1653𝐸−05	5.8904𝐸−04	20
Conical shell
𝜂=0.5: target frequency 𝑓1𝑇=268.753Hz and target displacement 𝑤max𝑇=𝑢3max𝑇=5.2639⋅10^−7m
MC	0.00984	−0.74086	14.77422	27.83370	330.94622	5.2073𝐸−07	1.9799𝐸−02	35
GA	0.08800	−0.32650	16.65690	7.72240	321.89420	5.2624𝐸−07	4.9477𝐸−04	25
PSO	0.05769	−0.49331	9.85753	8.35549	322.04900	5.2615𝐸−07	8.2088𝐸−04	43
Toroidal shell panel
𝜂=0: target displacement 𝑤max𝑇=𝑢3max𝑇=3.9516⋅10^−5m
MC	0.05350	−0.54251	17.60320	20.64836	20.21863	3.9727𝐸−05	5.3396𝐸−03	21
GA	0.02930	−0.78200	12.72730	25.90420	20.56750	3.9495𝐸−05	5.3143𝐸−04	3
PSO	0.10430	−0.58284	8.53859	8.22430	20.48775	3.9519𝐸−05	7.5919𝐸−05	6
Catenoidal panel
𝜂=1: target frequency 𝑓1𝑇=590.557Hz
MC	1.06122	0.90588	2.06831	2.50895	595.86858	3.9753𝐸−07	8.9937𝐸−03	55
GA	0.08800	−0.48390	13.11830	13.97850	590.77480	4.0135𝐸−07	3.6832𝐸−04	45
PSO	0.09089	−0.32061	18.33638	14.94431	590.11938	4.0129𝐸−07	7.4152𝐸−04	43
Elliptic paraboloid
𝜂=0.5: target frequency 𝑓1𝑇=327.239Hz and target displacement 𝑤max𝑇=𝑢3max𝑇=1.9073⋅10^−6m
MC	0.29917	0.02997	19.54311	2.65521	332.68799	1.8923𝐸−06	1.2258𝐸−02	43
GA	0.08800	−0.29910	3.01080	4.93650	327.62800	1.9075𝐸−06	6.4729𝐸−04	22
PSO	0.25269	−0.65120	13.49764	2.90612	327.39815	1.9065𝐸−06	4.5339𝐸−04	25