![]() In Part II of this paper, pressure drop measurements in the OUBER geometry will be used to assess the uniaxial and biaxial extensional rheometry of dilute polymeric solutions, in comparison to measurements made in planar extension using an optimized-shape cross-slot extensional rheometer (OSCER, Haward et al, Phys. The flow velocimetry confirms the accurate imposition of the desired and predicted flows, with pure extensional flow at an essentially uniform deformation rate being applied over a wide region around the stagnation point. Employing a viscous Newtonian fluid with a refractive index matched to that of the optically transparent microfluidic device, we conduct microtomographic-particle image velocimetry in order to resolve the flow field at low Reynolds number (< 0.1) in a substantial volume around the stagnation point. Fabrication of the geometry, which we name the optimized uniaxial and biaxial extensional rheometer (OUBER), is achieved with high precision at the microscale by selective laser-induced etching of a fused-silica substrate. Of the various numerically-generated geometries, one is selected as being most suitable for fabrication at the microscale, and numerical simulations with the Oldroyd-B and Phan-Thien and Tanner models confirm that the optimal flow fields in the chosen geometry are observed for both constant viscosity and shear thinning viscoelastic fluids. We present a numerical optimization of a "6-arm cross-slot" device, yielding several three-dimensional shapes of fluidic channels designed to impose close approximations to ideal uniaxial (or biaxial) stagnation point extensional flow under the constraints of having four inlets and two outlets (or two inlets and four outlets) and Newtonian creeping flow conditions. Financial support was provided by the UGC of Hong Kong under the Hong Kong Ph.D. Xiaojun Chen at The Hong Kong Polytechnic University. thesis finished under the supervision of Dr. These experiments demonstrate that COBYQA is an excellent successor to COBYLA as a general-purpose DFO solver. We expose extensive numerical experiments of COBYQA, showing evident advantages of COBYQA compared with Powell's DFO solvers. An important feature of COBYQA is that it always respects bound constraints, if any, which is motivated by applications where the objective function is undefined when bounds are violated. This derivative-free trust-region SQP method is designed to tackle nonlinearly constrained optimization problems that admit equality and inequality constraints. Finally, we elaborate on developing our new DFO method, named COBYQA after Constrained Optimization BY Quadratic Approximations. In particular, we show that the objective function of the SQP subproblem is a natural quadratic approximation of the original objective function in the tangent space of a surface. Therefore, we present an overview of the SQP method and provide some perspectives on its theory and practice. Moreover, a significant part of this thesis is devoted to developing a new DFO method based on the sequential quadratic programming (SQP) method. In particular, this thesis presents PDFO, a package we develop to provide both MATLAB and Python interfaces to Powell's model-based DFO solvers, namely COBYLA, UOBYQA, NEWUOA, BOBYQA, and LINCOA. These methods are motivated by optimization problems for which it is impossible or prohibitively expensive to access the first-order information of the objective function and possibly the constraint functions. This thesis studies derivative-free optimization (DFO), particularly model-based methods and software. The results indicated that the KSE generally took fewer function evaluations to find the global optima or reach the target value in most test problems, holding better efficiency and robustness. ![]() The KSE was compared to other global surrogate-based optimization methods on 12 bound-constrained testing functions with 2 to 16 design variables and 2 aerodynamic optimization problems with 24 to 77 design variables. By combining local and global searches, the proposed method could improve the fitting quality of the surrogate model and the optimization efficiency. During the local search, an enhanced trust-region method was adopted to make deep exploitation. ![]() Then, the partially explored minima would be furtherly exploited. ![]() It selected multiple promising local minima and classified them into partially and fully explored minima in terms of the fitting quality of the surrogate model. To this end, a Kriging-based global optimization method, named the Kriging-based space exploration method (KSE), was proposed in this paper. For complicated aerodynamic design problems, the efficient global optimization method suffered from the defect of the incorrect portrayal of the design space, resulting in bad global convergence and efficiency performance.
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