Scientific background | Structural Object Oriented Optimization

english українська [/ua/] русский Structural Object Oriented Optimization of Technical Systems Home Factory Software OptCAD Scientific background Our papers New approach Dissertations Contacts Scientific background 1. Optimization problem formulation 2. Mathematical apparatus 3. Structural optimization methodology 1. Optimization problem formulation We pose the structural optimization problem as the non-linear programming task: searching the values of variable structural parameters with minimizing some objective function subject to system of constraints. We propose that structural topology, support conditions, node connections of bars and pattern load are prescribed and constants. Mathematical model of optimization problem consists of design variable set, system of constraints and purpose function. The variable structural parameters (design variables) are: geometrical variables (node coordinates of the structural shape); cross-sectional variables (cross-sectional dimensions of bars); stresses (pre-stressing forces in redundant members). The specific technical-and-economic index can serve as the objective function. On the determining for purpose function, we take into account design specifications and ability to formulate the analytical expression as function of design variables. We enter an equilibrium conditions for each case of structural loading into the system of constraints. Beside this load-carrying ability and stiffness conditions for structural components and entire construction according to building regulations are included to system. Architectural, technological and other requirements might by integrate to constraint system also. For example, constraints for each cross-section type, which describe allowed value domain of it geometrical characters, number discrete range of rolled products and technological condition for welded elements are involved to mathematical model. 2. Mathematical apparatus In order to solving the optimization problem we use the algorithm of finite element method for structural analysis and method of non orthogonal projection of purpose function gradient for searching optimal design decision. Searching of optimal design decision is an iterative process of his modification building in the design variable surface, which ensures convergence to design decision with minimal value of objective function (optimum point). Proposed modification of the gradient method ensures the selection of linear-independent constraints with constant lengths of constraint and purpose function gradients. 3. Structural optimization methodology An algorithm for solving of the structural optimization problem consists of following steps: Step 1. Specification the start design decision and initial data for calculating. We describe design variables. The initial data for structural optimization problem are structural topology, material characteristics of it elements, types of bar cross-sections, support conditions, placements of hinges, pattern and values of the design load, pre-stressed diagram, deflection limits, optimality criteria and additional constraints. Step 2. Calculating of geometrical and effective bar lengths. Geometrical and effective bar lengths is determined using initial data. At the next iteration length values is corrected by current values of geometrical design variables. Step 3. Calculating of cross-sectional parameters. We determine geometrical characteristics (areas, moments of inertia, modulus of sections, static moments) and effective sizes of bar cross-sections uses continuous analytical dependence from values of cross-sectional design variables. Step 4. Static analysis. Using finite element method we calculate for each load case linear displacements for all structural nodes and stresses, which occurred in each design section for all structural bars. Step 5. Constraint inspection and forming of the set of active constraint numbers. We checking that the design structure comply with requirements (system of constraints) for all load cases and all design sections of structure. We identifying active (breached) constraints. Step 6. Calculating the current purpose function value and it gradient, determining the desire purpose function increment. The purpose function gradient is calculated by numerical differentiation using the finite difference approximation. We specifying the desire purpose function increment about 5...25% from current value of objective function. Step 7. Forming the vector of residuals and the matrix of gradients of active linear-independent constraints with triangular structure. Step 8. Calculating design variable increments and improved approximation of the optimum design decision according to gradient method equations. Step 9. Checking of stop conditions for the iteration search. If the current design decision not fulfils all constraints with assumed precision, we returning to step 2. Created by Max Sidorenko / 2008