Ulf Ringertz

ON METHODS FOR DISCRETE STRUCTURAL OPTIMIZATION

Abstract Three algorithms for discrete structural optimization are considered. The original discrete structural optimization problem is replaced by a sequence of approximate discrete subproblems. For small size problems a Branch and Bound method is used to find a global minimum for each subproblem. The use of a generalized Lagrangean function and nonconvex duality is compared to the use of the Branch and Bound method. The generalized Langrangean function is minimized over the discrete set using a neighbourhood search technique. The dual function is maximized using a subgradient method. A method for rounding off the continuous solution is also used. The performance of the algorithms is investigated on several numerical examples. Global discrete minima are found for two convex test problems by using Branch and Bound on the original discrete structural optimization problem.

On finding the optimal distribution of material properties

The compliance of two-dimensional structures is minimized using the elements of the constitutive matrix as design variables. The amount of material to be used is required to be less than a prescribed amount and further constraints on the constitutive matrix are imposed as matrix inequalities. The apparent nonsmoothness of the matrix inequalities is removed by use of a barrier transformation.

A BRANCH AND BOUND ALGORITHM FOR TOPOLOGY OPTIMIZATION OF TRUSS STRUCTURES

An algorithm for the selection of a minimum weight truss, out of a set of possible candidate trusses, is presented. The trusses are subject to stress and deflection constraints. Multiple loading conditions are considered. Starling with a ground structure, a sequence of subtrusses called candidate trusses are generated and analyzed. Several criteria are used to rapidly discard non-optimal configurations. Sequential Quadratic Programming is used to solve a non-linearly constrained problem which is part of the algorithm. A technique, frequently used to keep second derivative approximations positive definite, is found to give numerical instabilities. Possible modifications to improve stability are discussed. Finally, three examples are given to demonstrate the algorithm.

On structural optimization with aeroelasticity constraints

The optimal design of a cantilever wing in incompressible flow is considered. The wing is modelled as a full depth sandwich wing using finite element analysis. A doublet lattice panel method is used for computation of the unsteady aerodynamic loads. The weight of the wing is minimized using the thicknesses of the composite face sheets as design variables subject to constraints on flutter and divergence speed. Imperfection sensitivity of the final design is analysed and general aspects of imperfection sensitivity in optimization subject to aeroelasticity constraints are discussed in some detail.

Efficient Computation of Robust Flutter Boundaries Using the mu-k Method

A simple and efficient algorithm for robust flutter analysis is presented. First, a general linear fractional transformation formulation of the mu-k method is provided, making it straightforward to pose the uncertain flutter equation in a form suitable for structured singular value analysis. The new formulation establishes a close connection between mu-k flutter analysis and traditional frequency-domain flutter analysis, which is used to formulate an efficient algorithm for computation of robust flutter boundaries. The proposed method is successfully applied to an F-16 sample test case with uncertain external stores aerodynamics, showing that standard tools for structural dynamics and unsteady aerodynamics can be used to perform robust flutter analysis with only modest additional modeling.

EIGENVALUES IN OPTIMUM STRUCTURAL DESIGN

Eigenvalues frequently appear in structural analysis. The most common cases are vibration frequencies and eigenvalues in the form of load magnitudes in structural stability analysis. In structural design optimization, the eigenvalues may appear either as objective function or as constraint functions. For example maximizing the eigenvalue representing the load magnitude subject to a constraint on structural weight.

Lateral Stability and Control of a Tailless Aircraft Configuration

The aerodynamics of a generic aircraft configuration known as Swing was investigated. The configuration used had two trailing-edge flaps on each wing used for control around all three stability axes included pitch, roll, and yaw. Tests are done in the low-speed wind tunnel L2000 at a freestream velocity of 30 m/s corresponding to a Reynolds number of 6.9 · 105 was applied. A NACA-66009 airfoil was used for the wing, and the outer wing section was twisted up 5° around the leading-edge point of the wing. Asymmetric deflections with the same amplitude for both flaps on each wing were applied for roll control. The deflection of the flaps was recorded with a motion capture system. The infrared light was emitted from four cameras and the light was reflected back to the cameras with a scan rate of 100 Hz and the system computed the distance to the markers. The results revealed that a sudden increase in pitch moment were observed at angles of attack above 10° and the flaps deflected with the same amplitude.

Aeroelastic tailoring considering uncertainties in material properties

The robustness of aeroelastic design optimization with respect to uncertainties in material and structural properties is studied both numerically and experimentally. The model consists of thin orthotropic composite wings virtually without fuselage. Three different configurations with consistent geometry but varying orientation of the main stiffness axis of the material are investigated. The onset of aeroelastic instability, flutter, is predicted using finite element analysis and the doublet-lattice method for the unsteady aerodynamic forces. The numerical results are experimentally verified in a low-speed wind tunnel. The optimization problem is stated as to increase the critical air speed, above that of the bare wing by massbalancing. It is seen that the design goals are not met in the experiments due to uncertainties in the structural performance of the wings. The uncertainty in structural performance is quantified through numerous dynamic material tests. Once accounting for the uncertainties through a suggested reformulation of the optimization problem, the design goals are met also in practice. The investigation indicates that robust and reliable aeroelastic design optimization is achievable, but careful formulation of the optimization problem is essential.

Aeroelastic Design Optimization with Experimental Verification

This thesis treats various aspects of structural polymercomposites in aircraft applications. The mechanical performanceand quality of resin transfer molded (RTM) carbon fiberreinforced epoxy composites is studied. In a first part, the influence of manufacturing process parameters on the mechanicalbehavior of laminates is experimentally investigated. A number of process parameters are used as variables and performance ismeasured in terms of tensile and compressive strength as wellas interlaminar fracture toughness. The process parameters are concluded to have little affect on the measured properties. In a second part, the quality and structural performance of an entirely co-cured RTM manufactured aircraft control surfacedemonstrator is investigated. A series of quasi staticstructural tests using distributed loading is performed. Experimental results are compared with finite element analysis. Effects of impact damage on the performance are also studied.Good agreement is obtained between the predictions and the experiments. A nondestructive method for determination of elasticmaterial properties of orthotropic plates using naturalfrequencies is developed and verified. Finite elementcalculations of the natural frequencies of the plate are matched to experimentally determined frequencies using theelastic constants as variables. The method is successfully verified even for nontrivial specimen geometries with cornersingularities. Emphasis is on practical utilization ofknowledge about numerical and modeling errors as well asexperimental uncertainties. The optimal design of a thin orthotropic wing subject toaeroelastic constraints is studied using numerical methods andverified in low speed wind tunnel testing. The flutter speed ofthe wing is maximized using the laminate orientation asvariable. Further, the problem of increasing the flutter speed to a prescribed value using minimal amount of additional concentrated masses on a fixed wing design is investigated. The main objective of the study is to verify that the performance of the optimized design can be achieved also in experiments. It is found that the optimal design is very sensitive to uncertainties in material and structural properties.Consequently, this has to be accounted for in the problemformulation. It is shown, and experimentally verified, that the robustness requirements on the optimal design can be met byreformulating the optimization problem.

OPTIMIZATION OF STRUCTURES WITH NONLINEAR RESPONSE

Minimum weight design of structures with geometrically nonlinear behaviour is treated. The problem is formulated in two different ways. In the first formulation all the equilibrium equations are relaxed and treated as constraints. Both design variables and displacements are independent variables. In the second approach the displacements are transformed such that only a few of the equilibrium equations need to be treated as constraints in the optimization problem. The design variables and only a few transformed displacements are independent variables. The optimization problems associated with both formulations are solved numerically using a sequential quadratic programming (SQP) method. Truss structures are used to illustrate the performance of the SQP method when used to solve the two types of problems.