Abstract: Aerodynamic shape optimization plays a fundamental role in aircraft design. However, useful parameterizations of shapes for engineering models often result in high-dimensional design spaces which can create challenges for both local and global optimizers. In this paper, we employ an active subspace method (ASM) to discover and exploit low-dimensional, monotonic trends in the quantity of interest as a function of the design variables. The trend enables us to eciently and eectively nd an optimal design in appropriate areas of the design space. We apply this approach to the ONERA-M6 transonic wing, parameterized with 50 Free-Form Deformation (FFD) design variables. Given an initial set of 300 designs, the ASM discovered a low-dimensional linear subspace of the input space that explained the majority of the variability in the drag and lift coecients. This revealed a global trend that we exploited to nd an optimal design with reduced computational cost.

Abstract: This paper presents the design optimization of a mechanically decoupled six-axis force/torque (F/T) sensor by minimization of cross coupling error. The new term ‘principal coupling’ is proposed to define the largest cross coupling error. In the first design step of the F/T sensor, the locations of twenty-four strain gages in a sensor structure are predetermined, and four structural design variables are selected to be optimized. In the second step, an optimization framework that reduces principal coupling is developed. Multiple constraints on good isotropic measurement and safety are considered and formulated using the output strain of each strain gauge circuit. The optimal design utilizes FEM software and MATLAB interactively to perform effective shape optimization. As a result of shape optimization, principal coupling of a six-axis F/T sensor was reduced from 35% to 2.5% with good isotropy. The final design of the F/T sensor was fabricated for experimental verification and there was only 0.7% difference in principal coupling and 5.2% difference in the overall strain output between the numerical and experimental results. The optimal design results in this paper are expected to provide a design guideline for multi-axis F/T sensors with significantly reduced cross coupling error, one of the biggest technical obstacles in developing F/T sensors.

Abstract: Structural topology optimization.- On basic properties of Michell's structures.- Validation of numerical method by analytical benchmarks and verification of exact solutions by numerical methods.- Introduction to shape and topology optimization.- Homogenization method for shape and topology optimization.- Level set method for shape and topology optimization.- Compliance minimization of two-material elastic structures.- Some notes on topology optimization of vibrating continuum structures.- Topology optimization of vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps.- On optimum shape design and periodicity of band-gape beam structures.- Topological design for minimum dynamic compliance of continuum structures subjected to forced vibration.- Topological design for minimum sound emission from bi-material structures subjected to forced vibration.- Discrete material optimization of vibrating laminated composite plates for minimum sound emission.- Optimization of diffusive transport problems.- Fluid topology optimization: stokes and Navier-stokes models.- Topology optimization of coupled multi-physics problems.- Topology optimization based on the extended finite element method.- Topology optimization of meso- and nano-scale problems.- Topology optimization under uncertainty.- A brief review of numerical methods of structural topology optimization.- The free material design in planar elasticity.

Abstract: A level set method is used as a framework to study the effects of including material interface properties in the optimization of multi-phase elastic and thermoelastic structures. In contrast to previous approaches, the material properties do not have a discontinuous change across the interface that is often represented by a sharp geometric boundary between material regions. Instead, finite material interfaces with monotonic and non-monotonic property variations over a physically motivated interface zone are investigated. Numerical results are provided for several 2D problems including compliance and displacement minimization of structures composed of two and three materials. The results highlight the design performance changes attributed to the presence of the continuously graded material interface properties.

Abstract: We introduce the Deformable Simplicial Complex method to topology optimization as a way to represent the interface explicitly yet being able to handle topology changes. Topology changes are handled by a series of mesh operations, which also ensures a well-formed mesh. The same mesh is therefore used for both finite element calculations and shape representation. In addition, the approach unifies shape and topology optimization in a complementary optimization strategy. The shape is optimized on the basis of the gradient-based optimization algorithm MMA whereas holes are introduced using topological derivatives. The presented method is tested on two standard minimum compliance problems which demonstrates that it is both simple to apply, robust and efficient.

Abstract: The paper discusses the filtering of shape sensitivities as a mesh independent regularization method for very large problems of shape optimal design. The vertices of the simulation discretization grids are directly used as design morphing handles allowing for the largest possible design space. Still, however, there has been a lack of theory to consistently merging the sensitivity filtering into the standard optimization technology which is an ongoing topic of discussion in the community. The actual paper tries to overcome this burden. As a result it will be shown that there is a perfect transition between the sensitivity filtering and all the other shape parameterization techniques used for the shape optimization, as there are CAD-based techniques, subdivision surfaces or morphing box technologies. It appears that sensitivity filtering belongs to the most general and powerful control technologies available for shape optimal design. The success will be demonstrated by various illustrative examples which span from basic aspects to sophisticated applications in structural and fluid mechanics.

Abstract: Shape optimization based on the shape calculus is numerically mostly performed using steepest descent methods. This paper provides a novel framework for analyzing shape Newton optimization methods by exploiting a Riemannian perspective. A Riemannian shape Hessian is defined possessing often sought properties like symmetry and quadratic convergence for Newton optimization methods.

Abstract: This paper presents a numerical shape optimization method for the optimum free-form design of shell structures. It is assumed that the shell is varied in the out-of-plane direction to the surface to determine the optimal free-form. A compliance minimization problem subject to a volume constraint is treated here as an example of free-form design problem of shell structures. This problem is formulated as a distributed-parameter, or non-parametric, shape optimization problem. The shape gradient function and the optimality conditions are theoretically derived using the material derivative formulae, the Lagrange multiplier method and the adjoint variable method. The negative shape gradient function is applied to the shell surface as a fictitious distributed traction force to vary the shell. Mathematically, this method is a gradient method with a Laplacian smoother in the Hilbert space. Therefore, this shape variation makes it possible both to reduce the objective functional and to maintain the mesh regularity simultaneously. With this method, the optimal smooth curvature distribution of a shell structure can be determined without shape parameterization. The calculated results show the effectiveness of the proposed method for the optimum free-form design of shell structures.

Abstract: An antenna shape synthesis method is proposed that allows shaping of the antenna geometry prior to specification of the feed location and type. This reduces the constraints placed on the optimization process and can lead to potentially new designs due to the increased degree of freedom afforded. An appropriate feedpoint is easily chosen after shape optimization by selecting a location on the resulting structure for best impedance matching. The procedure is made possible through the use of characteristic mode concepts. Examples show that the antenna-Q values of the resulting shaped radiators closely approach the fundamental bounds.

Abstract: This paper presents a novel approach to design of the supersonic aircraft outer mold line (OML) by optimizing the A-weighted loudness of sonic boom signature predicted on the ground. The optimization process uses the sensitivity information obtained by coupling the discrete adjoint formulations for the augmented Burgers Equation and Computational Fluid Dynamics (CFD) equations. This coupled formulation links the loudness of the ground boom signature to the aircraft geometry thus allowing efficient shape optimization for the purpose of minimizing the impact of loudness. The accuracy of the adjoint-based sensitivities is verified against sensitivities obtained using an independent complex-variable approach. The adjoint based optimization methodology is applied to a configuration previously optimized using alternative state of the art optimization methods and produces additional loudness reduction. The results of the optimizations are reported and discussed.

Abstract: This paper proposes a two-step method for topology optimization, in which after global search is performed by the on-off method, and then local search is carried out by the level set approach. The genetic and gradient-based algorithms are employed for the former and latter optimizations, respectively. This present method is applied to two numerical examples; topology optimizations of a magnetic shield system and interior permanent magnetic motor. It is shown that this present method can find solutions with better performances in comparison with the conventional on-off method.

Abstract: The objective of this paper is to present constrained optimization results for a cold-formed steel (CFS) cross-section shape with maximum axial capacity. In the authors׳ previous work unconstrained shape optimization was performed via stochastic search and gradient-based algorithms. Unconstrained shape optimizations produced a significant capacity increase, more than 140%, above standard CFS cross-sections, but many of the solutions are highly unconventional and have potential limitations both with respect to end use (e.g. attaching boards for walls and floors) and cost of manufacturing. Column capacity is determined using the Direct Strength Method (DSM) which requires inputs for the local, distortional and global critical buckling loads. These critical loads are obtained using the finite strip method, as implemented in the open source software CUFSM, which allows essentially any potential cross-section to be evaluated. To advance the applicability of the optimized results, end-use constraints and manufacturing constraints on the number of rollers employed in forming were both successfully incorporated in the shape optimization presented in this paper, resulting in optimized cross-sections that are more practical and economical with only marginally decreased capacity (usually less than 10%) from the earlier unconstrained optimized solutions. The constraints are implemented within a simulated annealing (SA) algorithm for the optimization. Optimized sections from multiple runs show uniformity, partially indicating the robustness of the final optimized shapes. The implemented constrained shape optimization provides a thorough search with high computational efficiency. The optimized cross-sections from this research provide promising potential shapes for the development of new commercial product families, and the member-level optimization methodology can also be integrated into building optimization in the future.

Abstract: An adaptive geometry parametrization is presented to represent aerodynamic configurations during shape optimization. This geometry parametrization technique is constructed by integrating the classical B-spline formulation with a knot insertion algorithm. It is capable of inserting control points into a given parametrization without modifying the geometry. Taking advantage of this technique, a shape optimization problem can be solved as a sequence of optimizations from the basic parametrization to more refined parametrizations. Additional control points are inserted based on criteria incorporating sensitivity analysis and geometric constraints. Example problems involving airfoil optimization and induced drag minimization demonstrate the effectiveness of the proposed approach in comparison to uniformly refined parametrizations.

Abstract: The main contribution of this thesis is the implementation of manufacturing constraints in shape and topology optimization. Fabrication limitations related to the casting process are formulated as mathematical constraints and introduced in the optimization algorithm. In addition, based on the same theoretical and modelization tools, we propose a novel formulation for multi-phase optimization problems, which can be extended to the optimization of structures with functionally-graded properties. A key ingredient for the mathematical formulation of most problems throughout our work is the notion of the signed distance function to a domain. This work is divided into three parts. The rst part is bibliographical and contains the necessary background material for the understanding of the thesis' main core. It includes the rst two chapters. Chapter 1 provides a synopsis of shape and topology optimization methods and emphasizes the combination of shape sensitivity analysis and the level-set method for tracking a shape's boundary. In Chapter 2 we give a short description of the casting process, from which all our manufacturing constraints derive. We explain how industrial designers account for these limitations and propose a strategy to incorporate them in shape and topology optimization algorithms. The second part is about the mathematical formulation of manufacturing constraints. It starts with Chapter 3, where the control of thickness is discussed. Based on the signed distance function, we formulate three constraints to ensure a maximum and minimm feature size, as well as a minimal distance between structural members. Then, in Chapter 4, we propose ways to handle molding direction constraints and combine them with thickness constraints. Finally, a thermal constraint coming from the solidi cation of cast parts is treated in Chapter 5 using several thermal models. Multi-phase optimization is discussed in the third part. The general problem of shape and topology optimization using multiple phases is presented in detail in Chapter 6. A "smoothed-interface" approach, based again on the signed distance function, is proposed to avoid numerical di culties related to classical "sharp-interface" problems and a shape derivative is calculated. An extension of this novel formulation to general types of material properties' gradation is shown in the Appendix A.