On barrier and modified barrier multigrid methods for 3d topology optimization

One of the challenges encountered in optimization of mechanical structures, in particular in what is known as topology optimization, is the size of the problems, which can easily involve millions of variables. A basic example is the minimum compliance formulation of the variable thickness sheet (VTS) problem, which is equivalent to a convex problem. We propose to solve the VTS problem by the Penalty-Barrier Multiplier (PBM) method, introduced by R.\ Polyak and later studied by Ben-Tal and Zibulevsky and others. The most computationally expensive part of the algorithm is the solution of linear systems arising from the Newton method used to minimize a generalized augmented Lagrangian. We use a special structure of the Hessian of this Lagrangian to reduce the size of the linear system and to convert it to a form suitable for a standard multigrid method. This converted system is solved approximately by a multigrid preconditioned MINRES method. The proposed PBM algorithm is compared with the optimality criteria (OC) method and an interior point (IP) method, both using a similar iterative solver setup. We apply all three methods to different loading scenarios. In our experiments, the PBM method clearly outperforms the other methods in terms of computation time required to achieve a certain degree of accuracy

Reliability-based Topology Optimization of Trusses with Stochastic Stiffness

A new method is proposed for reliability-based topology optimization of truss structures with random geometric imperfections and material variability. Such imperfections and variability, which may result from manufacturing processes, are assumed to be small in relation to the truss dimensions and mean material properties and normally distributed. Extensive numerical evidence suggests that the trusses, when optimized in terms of a displacement-based demand metric, are characterized by randomness in the stiffness that follow the Gumbel distribution. Based on this observation, it was possible to derive analytical expressions for the structural reliability, enabling the formulation of a computationally efficient single-loop reliability-based topology optimization algorithm. Response statistics are estimated using a second-order perturbation expansion of the stiffness matrix and design sensitivities are derived so that they can be directly used by gradient-based optimizers. Several examples illustrate the accuracy of the perturbation expressions and the applicability of the method for developing optimal designs that meet target reliabilities

A modified particle swarm optimizer and its application to spatial truss topological optimization

p. 1044-1057Particle Swarm Optimization (PSO) is a new paradigm of Swarm Intelligence which is inspired by concepts from 'Social Psychology' and 'Artificial Life'. Essentially, PSO proposes that the co-operation of individuals promotes the evolution of the swarm. In terms of optimization, the hope would be to enhance the swarm's ability to search on a global scale so as to determine the global optimum in a fitness landscape. It has been empirically shown to perform well with regard to many different kinds of optimization problems. PSO is particularly a preferable candidate to solve highly nonlinear, non-convex and even discontinuous problems. In this paper, one enhanced version of PSO: Modified Lbest based PSO (LPSO) is proposed and applied to one of the most challenging fields of optimization -- truss topological optimization. Through a benchmark test and a spatial structural example, LPSO exhibited competitive performance due to improved global searching ability.Yang, B.; Bletzinger, K. (2009). A modified particle swarm optimizer and its application to spatial truss topological optimization. Editorial Universitat Politècnica de València. http://hdl.handle.net/10251/676

Development of specialized base primitives for meso-scale conforming truss structures

The advent of rapid manufacturing has enabled the realization of countless products that have heretofore been infeasible. From customized clear braces to jet fighter ducts and one-off dental implants, rapid manufacturing allows for increased design complexity and decreased manufacturing costs. The manufacturing capabilities of this process have evolved to the point that they have surpassed current design capabilities. Meso-scale lattice structures can now be built that contain more lattice struts than it is reasonable to efficiently define. This work has attempted to create a method for designing such lattice structures that is efficient enough to allow for the design of large or complex problems. The main hindrance to the design of complex meso-scale lattice problems is essentially the need to define the strut diameters. While it is obvious that a large design would contain more struts than can be specified by hand, designs also quickly surpass the current capabilities of computational optimization routines. To overcome this problem, a design method has been developed that uses a unit-cell library correlated to finite element analysis of the bounding geometry to tailor the structure to the anticipated loading conditions. The unit-cell library is a collection of base lattice primitives, or unit-cells, that have been specialized for certain applications. In this case, primitives have been created that perform best under the types of stress analyzed by finite element analysis. The effectiveness of this process has been demonstrated through several example problems. In all cases, the unit-cell library approach was able to create structures in less time than current methods. The resulting structures had structural performance slightly lower than similar models created through optimization methods, although the extent of this degradation was slight. The method developed in this work performs extremely well, and is able to create designs for even the most complex lattice structures. There is room for future development, however, in the streamlining of the design process and consideration of higher-order affects within unit-cells.M.S.Committee Chair: David, Rosen; Committee Member: Chris, Paredis; Committee Member: Seung-Kyum, Cho

Multiobjective robust shape and topology optimization of plane frames using order statistics

This paper presents a worst-case approach to robust optimization of plane frame structures under variation in uncertain parameters. The optimization procedure is first implemented without considering uncertainty, resulting in an optimal structure that may be unstable without bending stiffness. Based on such optimal solution, we then take variation in uncertain parameters into consideration and estimate the quantile response or trimmed mean of order statistics, where the quantile response is used as a relaxation of worst value of structural response. In order to obtain robust optimal solutions at various robustness levels, a multiobjective optimization problem is formulated and solved to simultaneously minimize the several order statistics or trimmed means with different orders. It is demonstrated in the numerical examples that the optimal distribution of cross-sectional areas of elements vary with the change of robustness level, and the convergence by using trimmed mean as estimation of quantile response is better than that of the simple order statistics

Mastering Uncertainty in Mechanical Engineering

This open access book reports on innovative methods, technologies and strategies for mastering uncertainty in technical systems. Despite the fact that current research on uncertainty is mainly focusing on uncertainty quantification and analysis, this book gives emphasis to innovative ways to master uncertainty in engineering design, production and product usage alike. It gathers authoritative contributions by more than 30 scientists reporting on years of research in the areas of engineering, applied mathematics and law, thus offering a timely, comprehensive and multidisciplinary account of theories and methods for quantifying data, model and structural uncertainty, and of fundamental strategies for mastering uncertainty. It covers key concepts such as robustness, flexibility and resilience in detail. All the described methods, technologies and strategies have been validated with the help of three technical systems, i.e. the Modular Active Spring-Damper System, the Active Air Spring and the 3D Servo Press, which have been in turn developed and tested during more than ten years of cooperative research. Overall, this book offers a timely, practice-oriented reference guide to graduate students, researchers and professionals dealing with uncertainty in the broad field of mechanical engineering

Optimal Design of Wire-and-Arc Additively Manufactured I-Beams for Prescribed Deflection

Alloys fabricated by wire-and-arc additive manufacturing (WAAM) exhibit a peculiar anisotropy in their elastic response. As shown by recent numerical investigations concerning the optimal design of WAAM-produced structural components, the printing direction remarkably affects the stiffness of the optimal layouts, as well as their shape. So far, single-plate specimens have been investigated. In this contribution, the optimal design of WAAM-produced I-beams is addressed assuming that a web plate and two flat flanges are printed and subsequently welded to assemble the structural component. A formulation of displacement-constrained topology optimization is implemented to design minimum weight specimens resorting to a simplified two-dimensional model of the I-beam. Comparisons are provided addressing solutions achieved by performing topology optimization with (i) conventional isotropic stainless steel and with (ii) WAAM-produced orthotropic stainless steel at prescribed printing orientations. Lightweight solutions arise whose specific shape depends on the selected material and the adopted printing direction

Parametric design and optimization of arched trusses under vertical and horizontal multi-load cases

This dissertation faces the problem of the optimum design of steel truss arches subject to multiple load cases. Arches are one of the most ancient shape-resistant structures, widely used in both civil engineering and architecture. For instance, arches can be considered as purely compressed structures, provided that their “line of thrust” coincides with the centre line of the arch. The “line of thrust” is the locus of the points of application of the thrusts (internal forces or stress resultants) that must be contained within the cross-section of the arch in such a way that the arch transfers loads to the foundations through axial compressive stresses only. As a matter of fact, the more the “line of thrust” differs from the centre line of the arch, the larger the unfavourable bending moments that arise in the arch. This is the reason why it is fundamental to pay close attention to the choice of the shape for an arch in order to minimize (or avoid when it is possible) unfavourable bending effects. Several analytical, graphical and physical methods are provided to find the optimal shape of a monolithic (single rib) arch subjected to a certain load case (i.e. the “funicular curve” for that load). However, if multiple load cases must be considered, it is not possible to find a proper optimal shape for an arch with single rib. In this case, the choice of truss arches with at least two chords becomes indispensable. Indeed, it has been demonstrated that structural optimization of in-plane truss arches with two chords subjected to a single load case leads to optimal solutions in which upper and lower chords tend to coincide with each other and with the “funicular curve” (i.e. the “line of thrust”) for that load. In light of the above, simultaneous shape and size optimization of steel truss arches with two arched chords linked each other through a bracing system (with variable Pratt-type pattern) has been performed for multiple load cases and different structural boundary conditions. Truss arches are effectively used in arch bridges, especially when the arch span exceeds 200 meters (five out of the six steel arch bridges with a span over 500 m are truss arch bridges). For this purpose, a hybrid optimization routine integrating a parametric definition of the design problem, a metaheuristic optimization algorithm and a code for Finite Element Analysis (FEA) has been developed through a MATLAB program. The proposed optimization method allows to simultaneously optimize a larger set of design variables, notwithstanding their large number and various nature (topology, shape and size, as well as continuous and discrete variables, have been concurrently considered). Third-degree Rational Bézier Curves have been chosen to optimize the shape of the arch chords because they can represent a wide family of curves (including conic curves), depending on a small number of parameters. In so doing, in-plane truss arches with different span lengths and structural boundary conditions have been optimized for multiple load cases, only considering vertical loads (acting on the same plane as the arch), since in-plane arches are not suited to withstand out-of-plane loads. On the other hand, spatial arched trusses with two arched chords lying on different planes have been optimally designed for multiple loadings acting in different directions. In particular, a steel arched truss with a lower arched chord variably inclined in the 3D-space and a horizontal upper arched chord linked each other through a bracing system has been designed and optimized for three vertical load cases and a horizontal seismic action parallel to the upper chord plane. Thus, analysing the obtained results, useful suggestions for steel truss arch design have been deduced and presented in this dissertation