Stojan Kravanja

The multilevel MINLP optimization approach to structural synthesis: the simultaneous topology, material, standard and rounded dimension optimization

The paper describes the simultaneous topology, material, standard and rounded dimension optimization of mechanical structures, performed by the Mixed-Integer Non-linear Programming (MINLP) approach. Beside the generation of an MINLP mechanical superstructure, the development of a general multilevel MINLP formulation for a mechanical superstructure is presented. The consideration of the discrete materials as well as standard and particularly rounded dimensions in structural synthesis significantly increases the combinatorics of the discrete optimization, which as a result may become too difficult to solve. A Linked Multilevel Hierarchical Strategy (LMHS) has been introduced for the solving of such large combinatorial problems. In order to decrease the effect of non-convexities, the Modified Outer-Approximation/Equality-Relaxation (OA/ER) algorithm has been applied. Four numerical examples of different complexities are presented to illustrate the proposed multilevel MINLP optimization approach: the optimization of two steel trusses, a composite I beam and a hydraulic steel roller gate Intake gate, erected in Aswan II, Egypt.

Optimization based comparison between composite I beams and composite trusses

Abstract The paper presents the economical comparison between composite welded I beams and composite trusses, designed from hollow sections. The comparison was made for simply supported beams for different spans and different loads. Composite I beams and composite trusses were designed in accordance with Eurocode 4 for the conditions of both ultimate and serviceability limit states. The aim of the comparison was to find out the spans and loads, at which each of the different structures shows its advantages. In order to carry out a precise comparison between the two different composite systems, we applied a structural optimization method rather than classical structural analysis. The optimization was performed by the nonlinear programming approach. The economic objective function of self-manufacturing costs was defined for the optimization. This way, the obtained design solutions of both systems were not only optimal, but the optimization also caused that the design, resistance, stress and deflection (in)equality constraints were activated at the border of the feasible space. Consequently, the conditions of both the ultimate and serviceability limit states were fully exploited and there was no reserve of resistance for each of the two different structures. At this point the comparison between results of composite I beams and composite trusses was performed. Some contour diagrams are also shown at the end of the paper, which should enable researchers and engineers to come to a decision about different composite structures in a faster and more efficient way.

Efficient Multilevel MINLP Strategies for Solving Large Combinatorial Problems in Engineering

This paper reports on the experience gained in solving large combinatorial problems by using the Outer-Approximation /Equality-Relaxation (OA/ER) algorithm in two multilevel MINLP strategies. The first one is a Linked Multilevel Hierarchical Strategy (LMHS) and the second one is a Reduced Integer Space (RIS) strategy. Both strategies are used to decompose the original MINLP problem in a hierarchical manner into several MINLP levels that are much easier to solve than the original one. While the first LMHS strategy can be applied to problems that contain only simple mixed-integer constraints, e.g. standard dimensions, the RIS strategy can be used to solve problems with more complex mixed-integer constraints, e.g. different design equations for alternative units. The LMHS strategy is rigorous and can solve convex problems to global optimal solutions. On the other hand, when the RIS strategy is applied for the solution of large combinatorial problems, the global optimality cannot be guaranteed, but very good solutions can be obtained. The synthesis problem of a roller steel gate for a hydroelectric power station with 19623 binary variables is presented to illustrate the LMHS strategy, whilst the synthesis problem of a heat exchanger network comprising different types of exchangers with 1782 binary variables is presented to present the RIS strategy.

Cost optimization of industrial steel building structures

The paper presents the simultaneous cost, topology and standard cross-section optimization of single-storey industrial steel building structures. The considered structures are consisted from main portal frames, which are mutually connected with purlins. The optimization is performed by the mixed-integer non-linear programming approach, MINLP. The MINLP superstructure of different structure/topology and standard cross-section alternatives has been generated and the MINLP optimization model of the structure has been developed. The defined cost objective function is subjected to the set of (in)equality constraints known from the structural analysis. Internal forces and deflections are calculated by the elastic first-order analysis constraints. The dimensioning constraints of steel members are defined in accordance with Eurocode 3. The modified outer-approximation/equality-relaxation (OA/ER) algorithm, a two-phase MINLP strategy and a special prescreening procedure of discrete alternatives are used for the optimization. A numerical example of the cost optimization of a single-storey industrial steel building is presented at the end of the paper.

Shape and discrete sizing optimization of timber trusses by considering of joint flexibility

The paper presents the shape and discrete sizing optimization of timber trusses with the consideration of joint flexibility. The optimization was performed by the Mixed-Integer Non-linear Programming (MINLP) approach. In the optimization model an economic objective function for minimizing the structures self-manufacturing costs was defined. The design conditions in accordance with Eurocode 5 were considered as optimization constraints. The internal forces and deflections were calculated by finite element analysis. The structural stiffness matrix was composed by considering fictiously decreased cross-sectional areas of all the flexibly connected elements. The cross-section dimensions and the number of fasteners were defined as discrete sizing variables, while the joint coordinates were considered as shape variables. The applicability of the proposed approach is demonstrated through some numerical examples, presented at the end of the paper.

Updated Lagrangian formulation for nonlinear stability analysis of thin-walled frames with semi-rigid connections

This paper presents a one-dimensional (1D) finite element formulation for the nonlinear stability analysis of framed structures with semi-rigid (SR) connections. By applying the updated Lagrangian incremental formulation and the nonlinear displacement field of thin-walled cross sections, the equilibrium equations of a straight beam element are first developed. Force recovering is performed according to the external stiffness approach. Material nonlinearity is introduced for an elastic-perfectly plastic material through the plastic hinge formation at finite element ends. To account for the SR connection behavior, a special transformation procedure is developed. The effectiveness of the numerical algorithm discussed is validated through the test problems.

MINLP OPTIMIZATION OF STEEL FRAMES

In this paper we deal with the topology and standard optimization of unbraced steel frames with rigid beam-to-column connections. The optimization has been performed by the Mixed-Integer Non-linear Programming (MINLP) approach. The MINLP performs a discrete topology and standard dimension optimization, while continuous parameters are simultaneously calculated inside the continuous space. As the discrete/continuous optimization problem of steel frames is non-convex and highly non-linear, the Modified Outer-Approximation/EqualityRelaxation (OA/ER) algorithm has been used for the optimization. Two practical examples with the results of the optimization are shown at the end of the paper.

2.5 – The MINLP Approach to Cost Optimization of Structures

The paper presents the Mixed-Integer Non-Linear Programming (MINLP) approach to cost optimization of structures. The MINLP is a combined discrete/continuous optimization technique, where the discrete binary 0-1 variables are defined for the optimization of the discrete topology, material and standard sizes alternatives and the continuous variables for the optimization of continuous parameters. An economic objective function of the manufacturing material and labour costs is subjected to structural analysis and dimensioning constraints. The Modified Outer-Approximation/Equality-Relaxation (OA/ER) algorithm is used for the optimization. The accompanied Linked Multilevel Hierarchical Strategy (LMHS) accelerates the convergence of the algorithm. Three examples of the cost optimization of steel structures are presented at the end of the paper.

Numerical Simulation of Large-Displacement Behaviour of Thin-Walled Frames Incorporating Joint Action

This paper is concerned with the development of a beam formulation for large-displacement analysis of thin-walled frames with various joint flexibility conditions. By applying the updated Lagrangian incremental formulation and the displacement field of thin-walled cross-sections, which accounts for restrained warping and the second-order displacement terms due to large rotations, the equilibrium equations of a straight beam element are firstly developed. To include the influence of flexible joint behaviour into the numerical model, a special transformation procedure is proposed.

3.5 – Topology, Shape and Standard Sizing Optimization of Trusses Using MINLP Optimization Approach

The paper presents the simultaneous topology, shape and discrete/standard sizing optimization of steel trusses using Mixed-Integer Non-linear Programming (MINLP) approach. The discrete/continuous non-convex and non-linear optimization problems are solved by the Modified OA/ER algorithm. Two types of objective functions are introduced for the optimization, namely a volume/mass objective function and an economic objective function. The design constraints are defined in accordance to Eurocode 3. A numerical example is presented at the end of the paper in order to show the suitability of the proposed approach and to emphasize the importance of selecting a proper objective function.