The powerful genetic algorithm optimization technique is augmented with an innovative “domain-trimming” modification. The resulting adaptive, high-performance technique is called Genetic Algorithm with Domain-Trimming (GADT). As a proof of concept, the GADT is applied to a widely used benchmark problem. The 10-dimensional truss optimization benchmark problem has well documented global and local minima. The GADT is shown to outperform several published solutions. Subsequently, the GADT is deployed onto three-dimensional structural design optimization for offshore wind turbine supporting structures. The design problem involves complex least-weight topology as well as member size optimizations. The GADT is applied to two popular design alternatives: tripod and quadropod jackets. The two versions of the optimization problem are nonlinearly constrained where the objective function is the material weight of the supporting truss. The considered design variables are the truss members end node coordinates, as well as the cross-sectional areas of the truss members, whereas the constraints are the maximum stresses in members and the maximum displacements of the nodes. These constraints are managed via dynamically modified, nonstationary penalty functions. The structures are subject to gravity, wind, wave, and earthquake loading conditions. The results show that the GADT method is superior in finding best discovered optimal solutions.
Offshore structures offer presumably the greatest level of sophistication and difficulty in the analysis, design, and construction stages. This is due to multitude of complexities introduced by the offshore locations. The entailed unique conditions such as underwater currents, surface waves and wind, and seabed soil can often become problematic quickly. Such conditions introduce additional factors to the design challenge, such as extreme temperatures and rough weather conditions, marine growth, fatigue, corrosion, marine vessel impact, and interaction with aeroelastic and hydrodynamic loads. These conditions need to be thoughtfully taken into consideration due to the long and short term effects that can compromise the structure. The intricacy of the design and analysis process multiplies when earthquake loads are factored in. However, it is recommended to account for all loads simultaneously except for seismic loads, based on the potential occurrence combinations [
The design solution selected from a range of viable options often imposes additional restraints to the structural design as it shapes the structure’s response to the applied loads. The size, type, and configuration of the members supporting offshore structure, for instance, may produce a significant influence on marine growth size, which poses a consequent impact on hydrodynamic drag forces. The resulting effects of nonlinearities associated with fluid-structure interaction may be too significant for a thoughtless dismissal in the design phase [
A common simplified approach to the complex process of offshore structural design is the use of uncoupled load effects [
Devising optimal designs that satisfy multiple performance criteria, such as minimizing cost and maximizing efficiency, is arguably one of the most influential factors in modern structural design. Member-level (local) optimization techniques are routinely used to identify design quantities for acceptable structural performance. Within this framework, several optimization techniques have been utilized in the past [
Many structural optimization problems involve problem-specific constraints applicable to the solutions limiting the feasible search space. In these types of problems, it is challenging to adapt traditional optimization techniques to handle constraints. One of the most popular constraint handling methods is by incorporating penalty functions due to the relative simplicity and ease of implementation.
Topology optimization was used by many researchers to generate alternative structural design concepts for benchmark wind turbine blades. In these studies, the focus is alternative structural layout for wind turbine blades with the aim of improving its design, minimizing weight, and, ultimately, wind energy cost reduction (e.g., [
The materials’ cost for offshore wind turbine supporting structures constitutes a considerable amount of overall cost. Potentially significant cost savings that result from optimized structural designs encourage the incorporation of efficient optimization techniques. However, determining the optimal structural shape and weight of the supporting system is not a trivial task. The complexity is increased due to the mathematical description of loads (aeroelastic [wind], hydrodynamic, and seismic), the many variables describing the geometry, the nonsmooth objective function, and the constraints that have to be satisfied. The above considerations have inspired the implementation of the efficient Genetic Algorithm with Domain-Trimming (GADT) optimization technique.
This study presents the implementation of the GADT to achieve superior optimization. The capabilities of the presented optimization tool are demonstrated on a well-established optimization benchmark problem known for being challenging with known global and multiple local minima. Finally, the GADT tool is used to solve two typical
Hydrodynamics [the kinematics of the water particles] is a very sophisticated branch of applied science that has been studied extensively. The motion of salty ocean water is a result of several causes, such as tidal effects that initiate underwater currents, thermal gradient effects, surface-level winds, and topology of the seabed especially for shallow waters (where the water depth is lesser than half the characteristic wavelength). All of the mentioned which are site dependent and their individual influences and their interactions would potentially vary significantly.
Several wave theories exist which span in sophistication and acceptance. Among the widely used theories are the following: linear or Airy wave theory, Stokes 2nd-order and higher order theories, and stream-function and cnoidal wave theories [
When wave loading is applied to an offshore structure, by means of Airy’s wave theory and Morison’s equation, gravity loads are assumed to be simultaneous. The analysis often accounts for the various nonlinearities introduced by the nonlinear hydrodynamic force (drag and lift). Typical representation of wave loads is based on the three parameters: wave period, wave height, and mean water depth (Figure
Schematic representation of offshore loads (a) sea current wave profile (adopted from [
In such mathematical procedures, wave and current kinematic fields are often modeled using 5th-order Stokes wave theory. Forces on individual structural elements are then generated using Morison’s equation. These forces account for the inertial effects through the mass coefficient
Specifics and more details of quantifying such forces can be found elsewhere [
Waves induce orbital particle motion through the matter in which they travel; such orbits are typically closed but may undergo minor forward drift resulting from wind-surface interaction effects. So, in essence, currents are produced due to wave kinetic energy. If a current coincides with a wave propagation path or direction, the wave length is typically elongated [
Atmospheric temperature and pressure gradients provide wind with ample kinetic energy. As obstacles are placed in the path of this kinetic energy, some or all of it is converted into potential energy due to path deflection or obstruction and discontinuation. The generated potential energy is manifested through differential pressures on the obstacle body. The resulting pressures are influenced by many factors such as the obstacle shape, orientation in space, and contact area, as well as the wind speed and angle of approach. Wind-induced forces are highly dynamic in nature, but often times, for design purposes, it is deemed appropriate to represent them by equivalent static pressures.
Land-based structures approaches can be adopted for representing wind loads on offshore structures with proper adjustments accounting for open ocean surface roughness (lower category of surface roughness). The lesser roughness produces lower levels of turbulence than would be produced on land; this is translated into a slower rate of variation with height. That is, at any given height, the storm conditions are more severe and produce higher wind speeds in offshore locations as compared to onshore. An illustrative schematic is provided in Figure
Variation of mean wind speed with height (adopted from [
As the ground shakes under a structure during a seismic event, inertial forces are generated within the structure. During that oscillatory vibrational movement, other forces are engaged as well, namely, dissipative forces (damping) and restoring forces. The interaction between all these forces to maintain dynamic equilibrium gives rise to complex structural response. Being an extreme event, seismic activity often involves high structural response demands where inelastic and nonlinear response is to be expected, thus adding to the already complex interactive loading situation in an offshore site. Luckily, significant seismic events are not very frequent and require relatively a long time to build up (store) strain energy. This warrants separate (nonsimultaneous) consideration of earthquake loads from other loads for offshore structures. At offshore sites, tidal waves may or may not require special attention depending on the specific site conditions. Tidal waves travel with great velocities at sea but with small amplitude (wave height) as large bodies of water are displaced during an earthquake. It is only as these waves approach the shoreline that they attain huge amplitudes due to the shallow depths of the seabed. These scenarios are considered on site-specific basis and require special seismic hazard studies prior to any design efforts. The authors are not aware of any specific tidal wave design codes for offshore structures.
When seismic loads are deemed significant to control the offshore structural design either partially or entirely, equivalent static representations are often utilized. This is very much similar to design considerations of on-land sites. If the offshore structure under consideration is irregular or complex in geometry, dynamic (linear or otherwise nonlinear) response needs to be properly modeled. The details of the used design code, design loads, design load combinations, and so forth are specified in Section
In the master’s thesis of the third author [
The three basic operations of a GA—reproduction, crossover, and mutation—are the tools improving each population’s fitness across generations/iterations. During reproduction, better fit designs are selected, copied, and placed into a pool allowing each design to mate and reproduce. The selection method used in this study is the roulette wheel due to its simplicity and popularity. One prominent advantage of the GA optimization is its inherent resilience against converging to local optima rather than global optima. GA’s resilience can also be sometimes a potential shortcoming as its “
This study features a new technique developed to tackle GA’s limitation, which involves first reducing the selected domain and then recommencing the GA algorithm to enhance the probability of selecting the optimal solution. To demonstrate, take the previous example of an optimum solution of 2 which is searched for in a domain of
Naturally, the possibility that the global optima may exist within the removed (trimmed) region of the solution space persists. In such cases, the GADT will converge to equivalent solution achieved by the typical GA algorithm, as trimming is queued only by the initial convergence of the GA process. Experimentation has demonstrated that the GADT technique will typically outperform conventional GA as will be shown herein with the offshore support structure examples.
Penalty functions are very useful in enabling the solution of constrained problems as unconstrained. The solutions that violate any number of constraints are penalized, which results in associating high objective function values to nonfeasible solutions. Of the two types of penalty functions, stationary (static) function generally exhibit an inferior performance to that of nonstationary (dynamic) function. A nonstationary penalty function is generally defined as follows:
Flow chart of general GADT optimization.
Flowchart of domain-trimming
While
Penalty function considered options for eliminating unfeasible solutions; the (
The GADT technique is tested against a classical global optimization problem, the
Figure
10-bar planar cantilever truss model.
Table
Optimization results for the 10-bar planar truss.
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Optimal cross-sectional areas | ||||||
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Variable (in2) (cm2) | This work | Kaveh and Talatahari [ |
Lee and Geem [ |
Schmit and Farshi [ |
Schmit and Miura [ |
Venkayya [ |
Dobbs and Nelson [ |
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23.764 | 23.052 | 23.25 | 24.29 | 23.55 | 25.19 | 25.81 |
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0.101 | 0.1 | 0.102 | 0.1 | 0.1 | 0.363 | 0.1 |
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25.033 | 25.601 | 25.73 | 23.35 | 25.29 | 25.42 | 27.23 |
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14.113 | 15.139 | 14.51 | 13.66 | 14.36 | 14.33 | 16.65 |
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0.106 | 0.1 | 0.1 | 0.1 | 0.1 | 0.417 | 0.1 |
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1.987 | 1.969 | 1.977 | 1.969 | 1.97 | 3.144 | 2.024 |
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12.888 | 12.206 | 12.61 | 12.54 | 12.81 | 14.61 | 14.22 |
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12.427 | 12.568 | 12.21 | 12.67 | 12.39 | 12.08 | 12.78 |
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0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.513 | 0.1 |
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20.253 | 20.33 | 20.36 | 21.97 | 20.34 | 20.26 | 22.14 |
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Max deflection | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 1.82 |
Max stress | 25.0 | 25.0 | 25.0 | 25.0 | 25.0 | 23.3 | 25.0 |
The optimal solutions found by GADT satisfied all the problem constraints and Table
Figure
Path to optimum solution for the 10-bar planar truss via GA and GADT.
The proposed optimization problem for the supporting wind turbine truss structure is highly sophisticated as it requires the determination of the optimum configuration of geometry, shape, and member sizes simultaneously. The mathematical expression of the problem is as follows:
Combining the penalty and objective functions produces the fitness function used in GADT:
To limit the spatial shapes of structure to practical and feasible potential solutions, joint coordinates are restricted to remain on the main (vertically inclined) legs but are free to move along them (see Figure
Description of the design groups (variables) in optimization problem.
The efficiency of the GADT algorithm is demonstrated through two widely used offshore wind turbine support systems, namely, the quatropod and the tripod jacket trusses (Figure
Wind turbine supporting truss structures: (a) quatropod jacket and (b) tripod jacket.
The design constraints included the traditional permissible member stresses and truss joint displacements prescribed in the adopted AISC-LRFD design standard. All structural elements are modeled as frame elements subjected to gravity as well as wind, wave, and seismic loads. The dead loads are accounted for by the member dimensions and density, while the turbine weight is superimposed at the appropriate location. Typical live loads are treated as movable and temporary, and, as such, a static equivalent uniform 50 psf (2.39 kPa) was used. A typical water depth for the type of jacket was adopted 110 ft (33.5 m). The wave loading application was carried out via a 5th-order Stokes theory representation. The design wave was designated as a 100-yr return event with a height of 26 ft (7.92 m) and the period of 8 sec. The horizontal velocity contours for the waves during the 100-yr design storm conditions are shown in Figure
Horizontal wave velocity contours for the 100-yr return storm event [
Seismic loads were accounted for using the 2006 International Building Code [
Earthquake load case definition [
The GADT algorithm is applied to the structures mentioned above. The effects of the GADT control parameters (population size, penalty coefficient, and exponent) on the optimum solutions are explored and the combination producing the best results is adopted but omitted for brevity.
The truss members are assigned to one of 14 member groups as the 64-bar 3D truss structure is symmetric about both principal axes (
Optimal design values for the 3D 64-bar space truss (quatropod jacket).
Optimal design variables (m) | ||||||||
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Variable | |
Variable | BDV | Variable | BDV | |||
1 |
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0.74 | 12 |
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0.14 | 23 |
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0.57 |
2 |
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0.055 | 13 |
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13.53 | 24 |
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0.08 |
3 |
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0.71 | 14 |
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0.96 | 25 |
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13.50 |
4 |
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0.047 | 15 |
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0.12 | 26 |
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8.00° |
5 |
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0.64 | 16 |
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11.75 | 27 |
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0.59 |
6 |
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0.036 | 17 |
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0.74 | 28 |
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0.07 |
7 |
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0.55 | 18 |
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0.13 | 29 |
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0.45 |
8 |
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0.037 | 19 |
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13.91 | 30 |
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0.07 |
9 |
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0.54 | 20 |
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0.59 | 31 |
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5.10 |
10 |
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0.035 | 21 |
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0.12 | 32 |
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1.80 |
11 |
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0.84 | 22 |
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9.80 | 33 |
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0.06 |
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The GADT algorithm produced an optimum weight of 3,965,364 kgf after completing 500 searches as shown in Figure
Convergence history for the minimum weight of the 64-bar 3D truss.
Similar to the quatropod jacket, the tripod jacket truss members are assigned to one of the 14 member groups that are illustrated in Figure
Optimal design values for the 3D 48-bar space truss (tripod jacket).
Optimal design variables (m) | ||||||||
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Variable | |
Variable | BDV | Variable | BDV | |||
1 |
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0.73 | 12 |
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0.14 | 23 |
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0.60 |
2 |
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0.03 | 13 |
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12.83 | 24 |
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0.13 |
3 |
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0.62 | 14 |
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0.86 | 25 |
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12.94 |
4 |
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0.025 | 15 |
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0.14 | 26 |
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9.58° |
5 |
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0.59 | 16 |
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13.08 | 27 |
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0.36 |
6 |
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0.041 | 17 |
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0.65 | 28 |
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0.08 |
7 |
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0.46 | 18 |
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0.14 | 29 |
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0.40 |
8 |
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0.032 | 19 |
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13.55 | 30 |
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0.09 |
9 |
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0.55 | 20 |
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0.61 | 31 |
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5.27 |
10 |
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0.061 | 21 |
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0.12 | 32 |
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1.55 |
11 |
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1.06 | 22 |
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9.94 | 33 |
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0.06 |
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The GADT algorithm produced an optimum weight of 3,901,710 kgf after completing 1000 searches as shown in Figure
Convergence history for the minimum weight of the 48-bar 3D truss.
The GADT is observed to outperform the conventional GA technique in terms of discovering optima consistently throughout the three problems presented in this paper. Typically, such improvements to optimal solutions come at a computational price. The domain-trimming and GA reinitiation are the main sources of additional computational cost. However, it can be reported that the GADT is still considered to be a rather computationally efficient technique. In comparison to the conventional GA, the GADT optimization algorithm offers improvements to the discovered optima, which far exceed the minute extra numerical burden. A head-to-head performance comparison on the 10-bar benchmark problem demonstrated the superiority of the GADT over conventional GA. A more thorough and conclusive comparison between the GADT versus conventional GA can be included in a future study.
The genetic algorithm optimization technique is augmented with a novel “domain-trimming” variation. The resulting superior optimization technique is called Genetic Algorithm with Domain-Trimming (GADT). Initially, the GADT performance is demonstrated by application to the widely used 10-dimensional truss benchmark optimization problem. The 10-bar truss has best discovered global and local minima that are published in the literature. The GADT is shown to be superior to several published solutions. Subsequently, the GADT is deployed onto the main focus of this work, which is three-dimensional structural design optimization for offshore wind turbine supporting structures. This design optimization problem is highly complex entailing least-weight topology and member size optimization. The GADT is applied to two popular alternatives for the offshore wind turbine support structures, namely, quadropod and tripod jacket trusses. This design optimization problem, with its two versions, is nonlinearly constrained where the objective function is the material weight of the supporting truss. The selected design variables are the truss members’ cross-sectional areas, as well as the truss members’ end node coordinates (which determine the members’ lengths). The maximum displacements of the nodes and the maximum stresses in members are the applied constraints. These constraints are managed via nonstationary, dynamically modified penalty functions. During analysis and design of the two jacket structures, they are subjected to gravity, wind, wave, and earthquake loading conditions. The results show that the (GADT) method is very efficient in finding the best discovered optimal solutions in both cases. The resulting savings in the optimally designed offshore support structures using the GADT far outweighs the necessary additional computational effort.
The authors declare that there are no conflicts of interest regarding the publication of this paper.