General Geometric Constraints Research Articles

Flexible Formulation of Spatial Integration Constraints in Aerodynamic Shape Optimization

In aircraft design, spatial integration limits aerodynamic and structural performance. While the outer mold line shape determines aircraft aerodynamic characteristics, aircraft systems and passengers or payloads must fit inside. Emerging categories of aircraft, such as electric aircraft, are likely to experience new and critical spatial integration challenges. Existing aerodynamic shape optimization tools use a narrowly defined set of geometric constraints. However, no known aerodynamic optimization framework can handle spatial integration constraints derived directly from three-dimensional (3-D) geometry. A general geometric constraint formulation based on triangulated 3-D geometry is proposed. The constraint consists of two metrics: the length of the intersection curve(s) between any two objects, and the Kreisselmeier–Steinhauser function aggregated minimum distance. The proposed implementation of the intersection and distance computations is fast and analytically differentiable with respect to geometric design variables, making it suitable for efficient gradient-based optimization. The constraint formulation is validated using three Reynolds-averaged Navier–Stokes-based aerodynamic shape optimization problems: a 2-D fairing design problem, a 3-D fairing design problem, and the design of an aeroshell to surround a human avatar. It is shown that this approach is robust and efficient, enabling the aerodynamic optimization of bodies containing objects with arbitrary shapes.

Necessary Nondomination Conditions in Set and Vector Optimization with Variable Ordering Structures

In this paper we study the concept of nondomination in problems of set and vector optimization with variable ordering structures, which reduces to Pareto efficiency when the ordering structure is constant/nonvariable. Based on advanced tools of variational analysis and generalized differentiation, we develop verifiable necessary conditions for nondominated points of sets and for nondominated solutions to vector optimization problems with general geometric constraints that are new in both finite and infinite dimensions. Many examples are provided to illustrate and highlight the major features of the obtained results.

Sufficient conditions for global weak Pareto solutions in multiobjective optimization

In this paper we derive new sufficient conditions for global weak Pareto solutions to set-valued optimization problems with general geometric constraints of the type $$\begin{aligned} \text{ maximize}\quad F(x) \quad \text{ subject} \text{ to}\quad x\in \Omega , \end{aligned}$$ where \(F: X\rightrightarrows Z\) is a set-valued mapping between Banach spaces with a partial order on \(Z\). Our main results are established by using advanced tools of variational analysis and generalized differentiation; in particular, the extremal principle and full generalized differential calculus for the subdifferential/coderivative constructions involved. Various consequences and refined versions are also considered for special classes of problems in vector optimization including those with Lipschitzian data, with convex data, with finitely many objectives, and with no constraints.

EXPERIMENTAL INVESTIGATION OF AN INCOMPRESSIBLE FLOW IN A LABYRINTH SEALS

The present work is carried out to reduce the leakage through the labyrinth seal by optimizing the tooth profile and operating clearances. The effects of pressure ratio, clearance, tooth width and height, and number of throttles upon the leakage through a straight-through labyrinth seal for in­compressible flow were experimentally investigated. Measurements in a developed test section over wide geometric factors and flow conditions were carried out. The experimental results indicated that the pressure ratio had significant effect on the flow coefficient. The flow coefficient decreased with increasing the pressure ratio. As the clearance increased the flow coefficient increased. However, the flow coe­fficient was increased as the clearance, became very small. The flow coefficient reached a maximum value at tooth width to clearance ratio of 2. The effect of tooth height on the flow coefficient was found negligible for tooth height to clearance ratio higher than 2. The flow coefficient decreased as the number of throttles increased, then attained constant value when the number of throttles became larger than seven. The observed trends of the results for incompressible fluid through straight-through labyrinth seals can be used to establish some general geometric constraints for minimizing the leakage.

Affine Reconstruction from Translational Motion under Various Autocalibration Constraints

In this paper algorithms for affine reconstruction from translational motion under various auto calibration constraints are presented. A general geometric constraint, expressed using the camera matrices, is derived and this constraint is used in a least square solution to the problem. Necessary and sufficient conditions for critical motions are derived and shown to depend on the knowledge of the intrinsic parameters of the camera. Experiments on simulated data are performed to evaluate the noise sensitivity of the algorithms and the reconstruction quality for motions close to being critical. An experiment is performed on real data to illustrate that the method works in practice.

Deployability Conditions for Curved and Flat, Polygonal and Trapezoidal Deployable Structures

Prefabricated, deployable space frames that exhibit self-standing and stress-free states in both the deployed and collapsed configurations are investigated in this paper. This type of deployable structures shows considerable advantages as compared to previous designs that either required external stabilizing or had members with residual stresses in the deployed configuration. Following previous developments for flat deployable structures consisting of units with regular-polygon planviews, this study deals with flat structures made of trapezoidals, and curved structures assembled from regular-polygonal units. First, the general geometric constraints and deployability conditions for these units are formulated, and a methodology for using these constraints as geometric design criteria is presented. Furthermore, additional conditions for the assemblage of single units into larger structures are given. Then, structural analysis issues for this type of structures are discussed. The necessity of nonlinear analysis during deployment is emphasized. Finally, the above geometric design procedures are demonstrated with specific examples.