Mechanical engineering - LIGHTWEIGHT DESIGN OF MECHANICAL STRUCTURES

1 - Optimization

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NOTES OF LIGTHWEIGTH DESIGN OF MECHANICAL STRUCTURE Chiara Moreschini, AA 2022/23 MODULE 1: OPTIMIZATION • A design meeting all the requirements is called feasible • Active constraint = inequality constraint that satisfies the related equality (=) à once the optimum solution has been reached, the active constraint is the one constraining the problem Inactive constraint = inequality constraint that satisfies the related strict inequality (>,=0 u=0 à inactive constraint à must have s>=0 So if s=0 the Lagrange multiplier must be non - negative; however this condition can become a non - positivity condition depending on the formulation of the problem (like in the Excel solver). When imposing s=0 or u=0, one of the following conditions (u>=0 or s> =0 respectively) could be not satisfied, so in that case the KKT theorem’s hypotheses are not satisfied and the correct set of solution is the other one • Post - optimality analysis = analysis of what happens when a constraint is relaxed or tightened (effect on the cost function) à constraint variation sensitivity theorem : the Lagrange multipliers are equal to the variation of the cost function (first derivative) in the neighbourhood of the optimum point, so they give the benefit of relaxing a constraint or th e penalty associated with tightening it (constraint with largest multiplier = constraint with the largest impact on the cost function) !" ∗ = − % & " ∗ ' " " − % ( # ∗ ) # # ' " , ) # à small variation of the constraints around the optimum point Largest & " ∗ / ( # ∗ à largest variation of the cost function when changing the constraint of ' " , ) # (largest dividend to relax the constraint or largest penalty to tighten it) à understand how we can modify the optimum point without violating the constraint • Search met hods: gradient - based methods (steepest - descent method, constrained descent method, Newton’s method, quadratic programming subproblems, …) or direct search methods. GRD = Generalized Reduced Gradient algorithm used by excel solver to solve non linear progra mming problems with inequality constraints, that are treated as equalities introducing slack variables à • With the FEM model verify the shear stress distribution that has been hypothized analytically and compare: Results of analytical model Results of FEM model Deflection Maximum shear stress Maximum normal stress