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General optimization (esa.30)

esa.30

Through General optimization the parametric models can be optimized. The user specifies “what-to-optimize” and the strategy (mathematical method). The program calculates variants of the original project and iterates to the final solution. All steps are shown in a table and the best solution is highlighted.

SCIA Engineer Optimizer is a cutting edge software tool for the overall optimization of civil engineering structures. It represents a combination of a widespread structural analysis software (SCIA Engineer) and a separate optimization engine (EOT – Engineering Optimization Tool). The two programs have been integrated together and offer a versatile and complete optimization solution for all types of civil engineering structures.

- (SCIA Engineer) is a comprehensive software package for analysis, design and checks of civil engineering structures. The integration of (SCIA Engineer) into the process of overall optimization is enabled by its above-standard features:
- Parametrisation of the model: direct (numerical) values of individual properties of entities in SCIA Engineer can be replaced by parameters. The parameters can be viewed and edited directly in SCIA Engineer or via an open communication interface.
- AutoDesign: automatic search for an optimal design for a particular structural entity – e.g. an optimal size of a steel cross-section or an optimal reinforcement in a concrete cross-section determined on the base of calculated internal forces.
- XML interface for communication with other applications.

- EOT is an optimization solver in which the user defines the objective function for the optimization, determines relations between the parameters and selects the suitable optimization method. The solver finds the optimal solution according to the user’s input, trying to finish the task with the minimum possible number of steps.

## Optimisation workflow

The optimisation process can clearly be seen in the flow chart below. Once all required input data are entered, i.e. the model of the analysed structure is defined, the search for the optimal solution runs fully automatically and no interaction with the user is required. For real-life problems several optimal solutions can be found. In such situations, it is up to the user to make the final decision.

### 1. Creation of the model and its parameterisation

The model of the analysed structure is created using standard SCIA Engineer tools and functions. The geometry, boundary conditions, loads, etc. are defined.

Parameters are assigned to the properties that can vary during the optimization. A parameter indicates that a particular property becomes variable and that the user defines its initial value and, if required, also the limits.

If suitable or needed, it is possible to specify also relations between individual parameters (e.g. the relation between the width and height of a cross-section).

### 2. Definition of the objective function and selection of the optimization method

The objective function defines what the optimisation goal is. This can be a price, weight, dimensions, position of a support, location of a load.

Furthermore, it is necessary to select one of the available optimisation methods. The selection of the method may affect the time needed for the solution of the sought-after result.

### 3. Optimization cycle

a) The optimization solver (EOT) generates the set of parameters used for the creation of particular variants of the model.

b) SCIA Engineer receives these parameters, runs the prescribed calculations, code-checks and, if required, also AutoDesign.

c) In the next step, EOT returns the results and evaluates whether to modify the parameters in order to get closer to the desired optimal solution.

d) This process is repeated until the optimum value is reached.

### 4. Evaluation of the optimal solution

As already stated, the optimization finds one or more optima. It is the user who has to compare the obtained solution and make the final decisions.

### 5. Final check

The final set of parameters that gives the optimum solution is then used to create the final variant of structure. All required types of calculation and checks that were not performed during the optimisation can be executed now.

## EOT Optimization methods

Several different methods have been implemented in the EOT optimization solver.

### Gradient method: Sequential quadratic programming (SQP)

Gradient methods are known to be very efficient for continuous optimization problems. These are suitable, for example, when searching for the optimal positions of nodes, supports, or the optimal geometry of cross-sections. These methods cannot be used for optimization tasks when working with discrete values, such as when selecting rolled profiles from a library or when determining the number of reinforcement bars, etc. Gradient methods can be very fast; on the other hand, convergence problems may occur in projects with a large number of parameters and in tasks with a complicated shape of gradients.

### Stochastic methods: Modified simulated annealing (MSA), Differential evolution (DE)

Simply said, stochastic methods search for the result by means of “trial-and-error” and the evaluation of these “trials”. This group contains methods that are also called genetic algorithms. Stochastic methods are the most stable, but on the other hand, the required calculation time is much higher compared to gradient methods.

### Heuristic methods: Nelder-Mead (N-M)

Heuristic methods share the properties of both gradient and stochastic methods. Their speed, as well as the stability, is somewhere in between stochastic and gradient methods as well as the stability.

The difference between individual methods is illustrated on the following example. The optimization task is to find such positions of three intermediate supports of the continuous beam that produces minimum bending moments (both hogging and sagging). The pictures show the “progress” of individual methods from the starting (user-defined) state to the final computed state. Each colour in the diagrams corresponds to one of the intermediate supports. It is clearly seen that each method has its “own way” how to find the solution. In this particular example the SQP method proved to be the fastest one, nevertheless, there is no general rule that would always say in advance which method is the best for which type of optimisation task.

## Practical examples - Concrete

### Price optimization of continuous reinforced concrete beam

The objective of this optimisation project was to get the minimum total price of a two-span reinforced concrete beam. The beam is subjected to permanent and variable line loads. The rectangular cross-section (C25/30) is reinforced by longitudinal bars and stirrups. Independent parameters were the dimensions of the cross-section, number and diameters of longitudinal reinforcement bars and the diameter and the distance of stirrups.

The total time of the whole optimization procedure was about 4 hours 30 minutes and 1150 iterations were run. The final reinforcement pattern is shown in the following table.

The optimization found the dimensions and reinforcement of the beam. In the picture the gradual decrease of the objective function can be followed. The reduction of the total price reached was about 11%.

### Optimization of tendons geometry of post-tensioned concrete bridge

Concrete bridge is 46.54m long, has three spans (14.0+17.0+14.0) and two edge crossbeams. Construction stages are taken into account with time effects (creep and shrinkage of concrete). Pre-stressing is introduced by means of 10 tendons of Ls15.5-1860 material. Three different tendon shapes are used (see the picture). The objective is to optimize the shape of the tendons with the aim to minimize the total area of cross-sections of tendons.

Concrete bridges have to satisfy many kinds of design checks (ULS, SLS, details etc.). But not all of them must be necessarily included in the optimization in order to prevent too time-consuming calculations. Therefore, only the check of allowable concrete stresses has been introduced as a constraint in the optimization. The method of modified simulated annealing has been used. This algorithm found several optima. These optima were manually analyzed and only some of them satisfied all of the checks required by the code (i.e. the checks that were not included in the optimization).

Initial state | Sol.2 | Sol. 6 | Sol. 7 | |
---|---|---|---|---|

A_{p,req}[mm^{2}] |
27300 | 23100 | 25200 | 27100 |

Save [%] | - | 15.4 | 7.7 | 4.4 |

Number of iterations | 770 |

Total time of optimization | 11h 56min 40s (55.8s per iteration) |

The comparison of the accepted optima is illustrated in table 2. Savings in pre-stressing steel are about 15% in the solution number 2.

The optimized layout of tendons is the following:

- 6 pcs of 17-strand tendon with geometry A
- 2 pcs of 9-strand tendon with geometry B
- 2 pcs of 17-strand tendon with geometry C

## Practical examples - Steel

### Shape optimisation of steel truss girder

The objective of this example was to find the optimal geometrical shape of the girder itself and of individual profiles in order to reach the minimum total mass of the whole structure. The structure is a symmetrical simply supported truss girder made of RHS profiles subjected to point loads acting in the nodes of the bottom chord. Independent variables were the positions of the nodes and cross-sections of the members.

Unity check for the original shape | Unity check for the optimized shape |

The original weight of the structure was 524kg and optimized was only 335kg. It means savings of the material of about 36%.

The best optimization method for this case seemed to be the Nelder-Mead method which reached the solution after 230 iterations.

### Minimum weight of steel frame hall

Optimization criteria are shown in the picture:

A typical steel hall frame of 30 m span consists of two columns and two rafters. The used I-cross sections are welded, made of steel grade S355. The section depth is variable along the elevation of the columns and rafters are with haunches. The objective was to determine the minimal mass of the structure in function of the applied loads, through the optimization of the column variable cross-sections and haunches on rafters.

The Sequential quadratic programming method reached the optimum after 360 iterations. Mass of the original structure was 2115 kg, the optimized one was about 1713 kg.

Global check and deformation for original shape | Values for optimized shape |

28/01/2015