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

Optimal Design of Trusses With Geometric Imperfections: Accounting for Global Instability

A topology optimization method is proposed for the design of trusses with random geometric imperfections due to fabrication errors. This method is a generalization of a previously developed perturbation approach to topology optimization under geometric uncertainties. The main novelty in the present paper is that the objective function includes the nonlinear effects of potential buckling due to misaligned structural members. Solutions are therefore dependent on the magnitude of applied loads and the direction of resulting internal member forces (whether they are compression or tension). Direct differentiation is used in the sensitivity analysis, and analytical expressions for the associated derivatives are derived in a form that is computationally efficient. A series of examples illustrate how the effects of geometric imperfections and buckling may have substantial influence on truss design. Monte Carlo simulation together with second-order elastic analysis is used to verify that solutions offer improved performance in the presence of geometric uncertainties

Optimal Design of Trusses With Geometric Imperfections

The present paper focuses on optimization of trusses that have randomness in geometry that may arise from fabrication errors. The analysis herein is a generalization of a perturbation approach to topology optimization under geometric uncertainties. The main novelty in the present paper is in the consideration of potential buckling due to misaligned structural members. The paper begins with a brief review of the aforementioned perturbation approach, then proceeds with the analysis of the nonlinear effects of geometric imperfection. The paper concludes with some numerical examples

Large-volume lava flow fields on Venus: Dimensions and morphology

Of all the volcanic features identified in Magellan images, by far the most extensive and really important are lava flow fields. Neglecting the widespread lava plains themselves, practically every C1-MIDR produced so far contains several or many discrete lava flow fields. These range in size from a few hundred square kilometers in area (like those fields associated with small volcanic edifices for example), through all sizes up to several hundred thousand square kilometers in extent (such as many rift related fields). Most of these are related to small, intermediate, or large-scale volcanic edifices, coronae, arachnoids, calderas, fields of small shields, and rift zones. An initial survey of 40 well-defined flow fields with areas greater than 50,000 sq km (an arbitrary bound) has been undertaken. Following Columbia River Basalt terminology, these have been termed great flow fields. This represents a working set of flow fields, chosen to cover a variety of morphologies, sources, locations, and characteristics. The initial survey is intended to highlight representative flow fields, and does not represent a statistical set. For each flow field, the location, total area, flow length, flow widths, estimated flow thicknesses, estimated volumes, topographic slope, altitude, backscatter, emissivity, morphology, and source has been noted. The flow fields range from about 50,000 sq km to over 2,500,000 sq km in area, with most being several hundred square kilometers in extent. Flow lengths measure between 140 and 2840 km, with the majority of flows being several hundred kilometers long. A few basic morphological types have been identified

Structural Topology Optimization: Moving Beyond Linear Elastic Design Objectives

Topology optimization is a systematic, free-form approach to the design of structures. It simultaneously optimizes material quantities and system connectivity, enabling the discovery of new, high-performance structural concepts. While powerful, this design freedom has a tendency to produce solutions that are unrealizable or impractical from a structural engineering perspective. Examples include overly complex topologies that are expensive to construct and ultra-slender subsystems that may be overly susceptible to imperfections. This paper summarizes recent tools developed by the authors capable of mitigating these shortcomings through consideration of (1) constructability, (2) nonlinear mechanics, and (3) uncertainties

Multiple-Material Topology Optimization of Compliant Mechanisms Created via Polyjet 3D Printing

Compliant mechanisms are able to transfer motion, force, and energy using a monolithic structure without discrete hinge elements. The geometric design freedoms and multi-material capability offered by the PolyJet 3D printing process enables the fabrication of compliant mechanisms with optimized topology. The inclusion of multiple materials in the topology optimization process has the potential to eliminate the narrow, weak, hinge-like sections that are often present in single-material compliant mechanisms. In this paper, the authors propose a design and fabrication process for the realization of 3-phase, multiple-material compliant mechanisms. The process is tested on a 2D compliant force inverter. Experimental and theoretical performance of the resulting 3-phase inverter is compared against a standard 2-phase design.Mechanical Engineerin

Revisiting element removal for density-based structural topology optimization with reintroduction by Heaviside projection

We present a strategy grounded in the element removal idea of Bruns and Tortorelli [1] and aimed at reducing computational cost and circumventing potential numerical instabilities of density-based topology optimization. The design variables and the relative densities are both represented on a fixed, uniform finite element grid, and linked through filtering and Heaviside projection. The regions in the analysis domain where the relative density is below a specified threshold are removed from the forward analysis and replaced by fictitious nodal boundary conditions. This brings a progressive cut of the computational cost as the optimization proceeds and helps to mitigate numerical instabilities associated with low-density regions. Removed regions can be readily reintroduced since all the design variables remain active and are modeled in the formal sensitivity analysis. A key feature of the proposed approach is that the Heaviside functions promote material reintroduction along the structural boundaries by amplifying the magnitude of the sensitivities inside the filter reach. Several 2D and 3D structural topology optimization examples are presented, including linear and nonlinear compliance minimization, the design of a force inverter, and frequency and buckling load maximization. The approach is shown to be effective at producing optimized designs equivalent or nearly equivalent to those obtained without the element removal, while providing remarkable computational savings

Improved Two-Phase Projection Topology Optimization

Abstract Projection-based algorithms for continuum topology optimization have received considerable attention in recent years due to their ability to control minimum length scale in a computationally efficient manner. This not only provides a means for imposing manufacturing length scale constraints, but also circumvents numerical instabilities of solution mesh dependence and checkerboard patterns. Standard radial projection, however, imposes length scale on only a single material phase, potentially allowing small-scale features in the second phase to develop. This may lead to sharp corners and/or very small holes when the solid (load-carrying) phase is projected, or one-node hinge chains when only the void phase is projected. Two-phase length scale control is therefore needed to prevent these potential design issues. Ideally, the designer would be able to impose different minimum length scales on both the structural (load-carrying) and void phases as required by the manufacturing process and/or application specifications. A previously proposed algorithm towards this goal required a design variable associated with each phase to be located at every design variable location, thereby doubling the number of design variables over standard topology optimization [2]. This work proposes a two-phase projection algorithm that remedies this shortcoming. Every design variable has the capability to project either the solid or the void phase, but nonlinear, design dependent weighting functions are created to prevent both phases from being projected. The functions are constructed intentionally to resemble level set methods, where the sign of the design variable dictates the feature to be projected. Despite this resemblance to level sets, the algorithm follows the material distribution approach with sensitivities computed via the adjoint method and MMA used as the gradient-based optimizer. The algorithm is demonstrated on benchmark minimum compliance and compliant inverter problems, and is shown to satisfy length scale constraints imposed on both phases. 2. Keywords: Topology Optimization, Projection Methods, Manufacturing Constraints, Length Scale, Heaviside Projection. Introduction Topology optimization is a design tool used for determining optimal distributions of material within a domain. System connectivity and feature shapes are optimized and thus, as the initial guess need not be informed, topology optimization is capable of generating new and unanticipated designs. It is well-known, however, that this may result in impractical solutions that are difficult to fabricate or construct, such as ultra slender structural features or small scale pore spaces. A key focus of this work is to improve manufacturability of topology-optimized designs by controlling the length scale of the topological features. The length scale is generally defined as the minimum radius or diameter of the material phase of concern. It is thus a physically meaningful quantity that can be selected by the designer based on fabrication process. The fabrication process also dictates the phase (or phases) on which the restriction is applied. For example, for topologies constructed by deposition processes, it is relevant to consider constraining the minimum length scale of the solid phase. Similarly, for designs that are manufactured by removing material, for example by milling, the manufacturability constraints should include minimum length scale and maximum curvature of the voids as dictated by the machine. Moreover, it is well established that controlling the length scale has the additional advantage that it circumvents numerical instabilities, such as checkerboard patterns and mesh dependency. Several methods for controlling the length scale of a topology optimization design exist ([1], [3]). Herein, the Heaviside Projection Method (HPM) [1] is used. HPM is capable of yielding 0-1 designs in which the minimum length scale is achieved naturally, without additional constraints. In HPM, the design variables are associated with a material phase and projected onto the finite element space by a Heaviside function. This mathematical operation is independent of the problem formulation and the governing physics. The projection is typically done radially and the projection radius is chosen as the prescribed minimum length scale. In its original form [1], the method projects a single phase onto the elements and

Topology Optimization of Cold-Formed Steel Deck Diaphragms with Irregularities

The objective of this paper is to explore the optimization of building roofs composed of bare cold-formed steel deck profiles when subjected to lateral demands such that the diaphragm response dominates the roof design considerations. Through variation in the deck profile, deck thickness, sidelap connectors, and structural connectors the in-plane shear stiffness and strength, that may be realized by a bare steel deck acting as a diaphragm, covers a significant range. In addition, although deck orientation is not typically varied within a roof – the profiled nature of a steel deck provides starkly different in-plane rigidities along and across the deck profile. Here we consider the application of topology optimization to aid in determining an optimal layout for a cold-formed steel deck roof. The topology optimization problem is formulated employing planar orthotropic elements for the roof deck and seeks to determine the maximum stiffness (i.e. minimum compliance) under an equivalent static in-plane lateral load subject to constraints. Constraints are placed on the basic roof element properties that are consistent with ranges of available deck and connections. The optimizer considers thickness of the planar elements, in essence a proxy for in-plane stiffness, and orientation of the planar elements. Conversion of the optimization results into a realizable steel deck roof is demonstrated. A series of examples are considered, including a rectangular roof, as well as plan irregularities including non-rectangular building shape, and roof cutouts. Significant future challenges remain and are briefly enumerated.The authors gratefully acknowledge the financial support funded by the American Iron and Steel Institute, the American Institute of Steel Construction, the Steel Deck Institute, the Metal Building Manufacturers Association, the Steel Joist Institute and the US National Science Foundation through grant CMMI-1562821. Acknowledging the ideas and contributions collaborators in the Steel Diaphragm Innovation Initiative (SDII) have provided throughout the work on this paper. The authors also thank Krister Svanberg for providing the MMA optimizer code. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or other sponsors