## Mixed Variable Formulations for Truss Topology Optimization

##### Mela, Kristo (2013)

Mela, Kristo

Tampere University of Technology

2013

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**Julkaisun pysyvä osoite on**

http://urn.fi/URN:ISBN:978-952-15-3116-3

##### Tiivistelmä

A study on formulating truss topology optimization problems using continuous and binary variables is presented. The ground structure approach, where members and nodes are allowed to vanish from an initial dense truss, is adopted. Member cross-sections are chosen from a discrete set of alternatives. The binary variables are used to determine the existence of members and nodes as well as the selection of a profile for the truss members. Normal forces of the members and nodal displacements are chosen as continuous variables. The equations of structural analysis are written as constraints of the optimization problem. Further constraints ensure that the truss is able to carry the loads and is kinematically stable. Member strength and buckling constraints are formulated according to the design rules of Eurocode 3.

The aim of the optimization problems considered is to find economical truss designs. The weight of the truss serves as the default criterion as it can be easily evaluated and it is related to the total cost. However, it is well-known that the actual minimum cost design can differ from the minimum weight truss. Therefore, a feature-based cost function is also devised for tubular plane trusses for cost optimization. For design situations, where the cost data is not available, a multicriterion optimization problem where weight is minimized simultaneously with the number of truss members, nodes and profiles used in the design, is formulated and Pareto optimal solutions are generated.

The proposed formulations lead to mixed-integer linear optimization problems. State-of-the-art software is employed to solve a set of benchmark problems that verify the formulations and demonstrate the effect of different constraints. Optimum topologies for Euler buckling and Eurocode 3 buckling constraints are compared. Topology optimization of a roof truss is presented as a case study. The conflict of weight and cost is studied in conjuction with the roof truss.

The aim of the optimization problems considered is to find economical truss designs. The weight of the truss serves as the default criterion as it can be easily evaluated and it is related to the total cost. However, it is well-known that the actual minimum cost design can differ from the minimum weight truss. Therefore, a feature-based cost function is also devised for tubular plane trusses for cost optimization. For design situations, where the cost data is not available, a multicriterion optimization problem where weight is minimized simultaneously with the number of truss members, nodes and profiles used in the design, is formulated and Pareto optimal solutions are generated.

The proposed formulations lead to mixed-integer linear optimization problems. State-of-the-art software is employed to solve a set of benchmark problems that verify the formulations and demonstrate the effect of different constraints. Optimum topologies for Euler buckling and Eurocode 3 buckling constraints are compared. Topology optimization of a roof truss is presented as a case study. The conflict of weight and cost is studied in conjuction with the roof truss.

##### Kokoelmat

- Väitöskirjat [3912]