Minimum Size Constraints Research Articles

Optimizing Connected Components Graph Partitioning With Minimum Size Constraints Using Integer Programming and Spectral Clustering Techniques

ABSTRACTIn this work, a graph partitioning problem in a fixed number of connected components is considered. Given an undirected graph with costs on the edges, the problem consists of partitioning the set of nodes into a fixed number of subsets with minimum size, where each subset induces a connected subgraph with minimal edge cost. The problem naturally surges in applications where connectivity is essential, such as cluster detection in social networks, political districting, sports team realignment, and energy distribution. Mixed Integer Programming formulations together with a variety of valid inequalities are demonstrated and computationally tested. An assisted column generation approach by spectral clustering is also proposed for this problem with additional valid inequalities. Finally, the methods are tested for several simulated instances, and computational results are discussed. Overall, the proposed column generation technique enhanced by spectral clustering offers a promising approach to solve clustering and partitioning problems.

Discrete topology optimization in augmented space: integrated element removal for minimum size and mesh sensitivity control

A computational strategy is proposed to circumvent some of the major issues that arise in the classical threshold-based approach to discrete topology optimization. These include the lack of an integrated element removal strategy to prevent the emergence of hair-like elements, the inability to effectively enforce a minimum member size of arbitrary magnitude, and high sensitivity of the final solution to the choice of ground structure. The proposed strategy draws upon the ideas used to arrive at mesh-independent solutions in continuum topology optimization and enables efficient imposition of a minimum size constraint onto the set of non vanishing elements. This is achieved via augmenting the design variables by a set of auxiliary variables, called existence variables, that not only prove very effective in addressing the aforementioned issues but also bring in a set of added benefits such as better convergence and complexity control. 2D and 3D examples from truss-like structures are presented to demonstrate the superiority of the proposed approach over the classical approach to discrete topology optimization.

A hybrid density/level set formulation for topology optimization of functionally graded lattice structures

Additive manufacturing is capable of producing lightweight structures by introducing mesoscale lattice structures in the design without significant additional manufacturing costs. Nevertheless, designing efficient structures including complex lattices remains a challenging task. This paper presents a strategy for the design of additively manufactured structures that is able to simultaneously optimize the shape of the macroscale structure and the functionally graded lattices inside this design. Assuming a separation of scales between the lattice cell size and the macroscale structure, an effective material model based on numerical homogenization is adopted to model the lattice behavior in the analysis of the macroscale structure. While this material model is parametrized by volume densities to mimic functionally graded lattices, the shape of the macroscale structure is represented by a level set function and modeled by the extended finite element method inside the design domain. By adopting an explicit approach to the level set optimization the design problem can be formulated as a single nonlinear programming problem which can be solved with standard optimization algorithms for topology optimization. The proposed formulation avoids degenerate lattice members by excluding small densities in the optimization. Furthermore, the distinct representation of the macroscale structure enables to easily include geometric constraints on the macroscale level such as a minimum feature size or other constraints specific to the additive manufacturing process. The effectiveness of the method is demonstrated in a number of examples.

Integrated Methodology for Designing Structures coming from Additive Layer Manufacturing

The potential of Additive Layer Manufacturing (ALM) is high, with a whole new set of manufacture parts with unseen complexity being offered. However, the process has limitations such as minimum member sizes or overhang constraints, which should be taken into account in the design. Therefore, the combination of Topology Optimization (TO) with ALM can be seen as an advantageous one [1]. Nevertheless, the connection between TO and ALM represents a non-trivial step that requires a robust methodology. In this sense, a proposal towards an integrated robust methodology is shown and demonstrated in the present paper. The TO step is available and accounts for minimum member size (Heaviside projection) and/or overhang constraint (layer-wise simplified fabrication model) [2]. The algorithm uses a regular cubic approach with precomputed stiffness matrices for efficient assembling and solving, where the mentioned transition between TO and ALM is also available via Laplacian smoothing. The source code is available on GitHub in the Trimesh module, as all the other auxiliary tools, namely geometry import and export capabilities, visualization capabilities and interactive boundary conditions selection [3]. Finally, a 3D case-study is shown as a way to illustrate the potential of the prosed integrated process.

Providing open forest structural characteristics for high conservation priority wildlife species in southeastern U.S. pine plantations

The decline of early successional and open forests and their wildlife inhabitants has resulted in increased efforts to understand and conserve these communities. Natural and anthropogenic disturbances that historically created and maintained open forest conditions have been disrupted, necessitating wildlife use of alternative habitat sources (e.g., pine plantations, utility rights-of way, abandoned agricultural fields) to carry out their life history. We reviewed available literature to estimate the range of structural conditions suitable for four open forest species in regional decline and compared these ranges to available structure in loblolly pine (Pinus taeda) forests managed for economic return. Species included Bachman’s sparrow (Aimophila aestivalis), northern bobwhite (Colinus virginianus), prairie warbler (Dendroica discolor), and eastern diamondback rattlesnake (Crotalus adamanteus). We estimated habitat availability and connectivity for each of these four species during a 60-year simulation on an economically- and operationally-feasible pine landscape (>22,000 ha) managed for sawtimber production in the West Gulf Coastal Plain. Habitat requirements, including minimum patch size and dispersal constraints, of Bachman’s sparrow, northern bobwhite, prairie warbler, and eastern diamondback rattlesnake are generally met by pine stands in stand establishment and thinned, mid-rotation stages. However, habitat availability in pine plantations may be highly ephemeral, tends to occupy the upper end of basal area and canopy closure tolerance, and may be more suitable for open woodland species (e.g., Bachman’s sparrow) than grassland species (e.g., northern bobwhite). The range of habitat associations in this literature review highlights the need to refine targets of structural metrics identified by open pine restoration initiatives to encompass the full range of conditions occupied by open forest species. Current literature is strongly biased to avian habitat associations. Few papers explore habitat associations of herpetofauna of conservation concern in the southeastern U.S., and these species tend to be less mobile and therefore, more vulnerable to landscape changes and the ephemeral nature of open vegetative structure in pine plantations.

Efficient pixel-by-pixel optimization of photonic devices utilizing the Dyson’s equation in a Green’s function formalism: Part II Implementation using standard electromagnetic solvers

In part I of this series of papers, we introduced an efficient pixel-by-pixel optimization technique, utilizing the Dyson’s equation in a Green’s function formalism. This technique directly incorporates material and minimum feature size constraints in practical photonic devices. The implementation in part I, however, requires full storage of the Green’s function, which is costly. In this paper, we further develop this optimization technique by showing that only the diagonal part of the Green’s function needs to be stored, and by showing that the optimization process can be implemented using standard commercially available electromagnetic solvers. The development here should enable a wider use of this optimization technique.

STRUCTURAL SEGMENTATION OF POINT CLOUDS WITH VARYING DENSITY BASED ON MULTI-SIZE SUPERVOXELS

Abstract. Ground objects can be regarded as a combination of structures of different geometries. Generally, the structural geometries can be grouped into linear, planar and scatter shapes. A good segmentation of objects into different structures can help to interpret the scanned scenes and provide essential clues for subsequent semantic interpretation. This is particularly true for the terrestrial static and mobile laser scanning data, where the geometric structures of objects are presented in detail due to the close scanning distances. In consideration of the large data volume and the large variation in point density of such point clouds, this paper presents a structural segmentation method of point clouds to efficiently decompose the ground objects into different structural components based on supervoxels of multiple sizes. First, supervoxels are generated with sizes adaptive to the point density with minimum occupied points and minimum size constraints. Then, the multi-size supervoxels are clustered into different components based on a structural labelling result obtained via Markov random field. Two datasets including terrestrial and mobile laser scanning point clouds were used to evaluate the performance of the proposed method. The results indicate that the proposed method can effectively and efficiently classify the point clouds into structurally meaningful segments with overall accuracies higher than 96%, even with largely varying point density.

Note on spatial gradient operators and gradient-based minimum length constraints in SIMP topology optimization

Spatial gradient information of density field in SIMP (solid isotropic material with penalization) topology optimization is very useful for imposing overhang angle and minimum length (size) manufacturing constraints or achieving shell-infill optimization. However, the computation of density gradient is an approximation since the design space is discretized. There are several operators for this purpose, which arise from the image processing field. This note compares different gradient operators in the context of SIMP topology optimization method and suggests a new computation strategy to improve the accuracy of gradient estimation. We take a case study of spatial gradient-based minimum size constraints. New structural indicator functions are proposed to improve the general applicability of previous gradient-based minimum length constraints. This study is carried out in 2D structure examples to validate the methodology.

Mesoscale Electrochemical Performance Simulation of 3D Interpenetrating Lithium-Ion Battery Electrodes

Advancements in micro-scale additive manufacturing techniques have made it possible to fabricate intricate architectures including 3D interpenetrating electrode microstructures. A mesoscale electrochemical lithium-ion battery model is presented and implemented in the PETSc software framework using a finite volume scheme. The model is used to investigate interpenetrating 3D electrode architectures that offer potential energy density and power density improvements over traditional particle bed battery geometries. Using the computational model, a variety of battery electrode geometries are simulated and compared across various battery discharge rates and length scales to quantify performance trends and investigate geometrical factors that improve battery performance. The energy density vs. power density relationship of the electrode microstructures are compared in several ways, including a uniform surface area to volume ratio comparison as well as a comparison requiring a minimum manufacturable feature size. Significant performance improvements over traditional particle-bed electrode designs are predicted, and electrode microarchitectures derived from minimal surfaces are shown to be superior under a minimum feature size constraint, especially when subjected to high discharge currents. An average Thiele modulus formulation is presented as a back-of-the-envelope calculation to predict the performance trends of microbattery electrode geometries.

Auxetic structure design using compliant mechanisms: A topology optimization approach with polygonal finite elements

Compliant mechanisms are monolithic structures where the movement is given by the flexibility of the structure rather than the presence of joints and pins. The absence of joints allows the construction of compliant mechanisms in microscale. In this work, the compliant mechanism is designed by using topology optimization to generate microstructure unit cells that simulate the effect of auxetic materials, i.e. those with negative Poisson's ratio. Polygonal finite element meshes are introduced in the topology optimization formulation to avoid the hinges (one-node connections) in the compliant mechanism design. A pattern repetition constraint is applied to generate auxetic macrostructures. An integrated approach that combines a projection technique with a mapping technique is adopted to include the minimum member size constraint, making the topology optimization results possible to be manufactured. The connection of the topology optimization approach with additive manufacturing is demonstrated using 3D printers based on FFF (fused filament fabrication) and PolyJet technologies. Thus, computational simulation in connection with rapid prototyping are carried out to verify the results.

A novel optimization design method of additive manufacturing oriented porous structures and experimental validation

The development and application of inhomogeneous porous structures have greatly benefitted from the emergence of additive manufacturing (AM) techniques. At the same time, the inherent constraints of AM techniques have highlighted the importance of design for AM. In this paper, a novel method is proposed to optimize the design of AM oriented porous structures and improve their structural performance. In particular, this method optimizes porous structures using embedded voids and incorporates two novel filters, one of which satisfies the minimum feature size constraint while the other facilitates a support-free manufacturing process. The performance of the method was validated via numerical analyses and experiments, the results of which confirmed the excellent stiffness properties and adaptability to the AM constraints of the designed porous structure.

Subwavelength engineering and asymmetry: two efficient tools for sub-nanometer-bandwidth silicon Bragg filters.

Bragg filters stand as key building blocks of the silicon-on-insulator (SOI) photonics platform, allowing the implementation of advanced on-chip signal manipulation. However, achieving narrowband Bragg filters with large rejection levels is often hindered by fabrication constraints and imperfections. Here, we show that the combination of single-side corrugation asymmetry and subwavelength engineering provides a narrowband response with large corrugations, overcoming minimum feature size constraints of conventional Si Bragg filters. We comprehensively study the impact of the corrugation asymmetry in conventional and subwavelength single-etched SOI Bragg filters, showing their potential for bandwidth reduction. Finally, we experimentally demonstrate novel subwavelength geometry based on shifted corrugation teeth, achieving null-to-null bandwidths and rejections of 0.8nm and 40dB for the symmetric configuration and 0.6nm and 15dB for the asymmetric case.

Modeling of piezoelectric sensors adhesively bonded on trusses using a mathematical programming approach

In this study, piezoelectric sensors design adhesively bonded on truss elements is treated in the framework of mathematical programming. A numerical formulation based on the strength capacity of set structure, adhesive and piezoelectric sensor is proposed. Inside the formulations maximum strength capacity of the adhesive is considered as a limit value in the design. Two formulations are established to obtain the maximum strength of the set; the first one is built on the basis of finite differences and the other one on a formulation of finite elements both based on an admissible static field. The lower bound method applied to limit analysis is extended in this research to analyze trusses with sensors including the adhesive interface. Four examples are designed to assess the numerical methodologies in which the results are compared with other known data. The main contribution of this work is focused on finding the maximum coupling load that a piezoelectric sensor can read before being debonded based on the minimum size constraint of the sensor.

Clustering and Community Detection With Imbalanced Clusters

Spectral clustering methods which are frequently used in clustering and community detection applications are sensitive to the specific graph constructions particularly when imbalanced clusters are present. We show that ratio cut (RCut) or normalized cut (NCut) objectives are not tailored to imbalanced cluster sizes since they tend to emphasize cut sizes over cut values. We propose a graph partitioning problem that seeks minimum cut partitions under minimum size constraints on partitions to deal with imbalanced cluster sizes. Our approach parameterizes a family of graphs by adaptively modulating node degrees on a fixed node set, yielding a set of parameter dependent cuts reflecting varying levels of imbalance. The solution to our problem is then obtained by optimizing over these parameters. We present rigorous limit cut analysis results to justify our approach and demonstrate the superiority of our method through experiments on synthetic and real datasets for data clustering, semi-supervised learning and community detection.

Spatial, socio-economic, and ecological implications of incorporating minimum size constraints in marine protected area network design.

Marine protected areas (MPAs) are the cornerstone of most marine conservation strategies, but the effectiveness of each one partly depends on its size and distance to other MPAs in a network. Despite this, current recommendations on ideal MPA size and spacing vary widely, and data are lacking on how these constraints might influence the overall spatial characteristics, socio-economic impacts, and connectivity of the resultant MPA networks. To address this problem, we tested the impact of applying different MPA size constraints in English waters. We used the Marxan spatial prioritization software to identify a network of MPAs that met conservation feature targets, whilst minimizing impacts on fisheries; modified the Marxan outputs with the MinPatch software to ensure each MPA met a minimum size; and used existing data on the dispersal distances of a range of species found in English waters to investigate the likely impacts of such spatial constraints on the region's biodiversity. Increasing MPA size had little effect on total network area or the location of priority areas, but as MPA size increased, fishing opportunity cost to stakeholders increased. In addition, as MPA size increased, the number of closely connected sets of MPAs in networks and the average distance between neighboring MPAs decreased, which consequently increased the proportion of the planning region that was isolated from all MPAs. These results suggest networks containing large MPAs would be more viable for the majority of the region's species that have small dispersal distances, but dispersal between MPA sets and spill-over of individuals into unprotected areas would be reduced. These findings highlight the importance of testing the impact of applying different MPA size constraints because there are clear trade-offs that result from the interaction of size, number, and distribution of MPAs in a network.

Simulation and experimental validation of powertrain mounting bracket design obtained from multi-objective topology optimization

A framework of multi-objective topology optimization for vehicle powertrain mounting bracket design with consideration of multiple static and dynamic loading conditions is developed in this article. Incorporating into the simplified isotropic material with penalization model, compromise programming method is employed to describe the multi-objective and multi-stiffness topology optimization under static loading conditions, whereas mean eigenvalue formulation is proposed to analyze vibration optimization. To yield well-behaved optimal topologies, minimum member size and draw constraint are settled for meeting manufacturing feasibility requirements. The ultimate mounting bracket is reconstructed based on the optimum results. Numerical analyses of the bracket are performed, followed by physical tests. It is proven that topology optimization methodology is promising and effective for vehicle component design.

Gate sizing using geometric programming

A two-step transistor sizing optimization method based on geometric programming for delay/area minimization is presented. In the first step, Elmore delay is minimized using only minimum and maximum transistor size constraints. In the second step, the minimized delay found in the previous step is used as a constraint for area minimization. In this way, our method can target simultaneously both delay and area reduction. Moreover, by relaxing the minimized delay, one may further reduce area with small delay penalty. Gate sizing may be accomplished through transistor sizing tying each transistor inside a cell to a same scale factor. This reduces the solution space, but also improves runtime as less variables are necessary. To analyze this tradeoff between execution time and solution quality a comparison between gate sizing and transistor sizing is presented. In order to qualify our approach, the ISCAS’85 benchmark circuits are mapped to a 45 nm technology using a typical standard cell library. Gate sizing and transistor sizing are performed considering delay minimization. Gate sizing is able to reduce delay in 21 %, on average, for the same area and power values of the sizing provided by standard-cells library. Then, the transistor sizing is executed and delay can be reduced in 40.4 % and power consumption in 2.9 %, on average, compared to gate sizing. However, the transistor sizing takes about 23 times longer to be computed, on average, using a number of variables twice higher than gate sizing. Gate sizing optimizing area is executed considering a delay constraint. Three delay constraints are considered, the minimum delay given by delay optimization and delay 1 and 5 % higher than minimum delay. An energy/delay gain (EDG) metric is used to quantify the most efficient tradeoff. Considering the minimum delay, area (power) is reduced in 28.2 %, on average. Relaxing delay by just 1 %, area (power) is reduced in 41.7 % and the EDG metric is 41.7. Area can be reduced in 51 %, on average, relaxing delay by 5 % and EDG metric is 10.2.

An Object-Based Approach to Assessing the Organization of Tropical Convection

Abstract The organization of tropical convection is assessed through an object-based analysis of satellite brightness temperature data Tb, a proxy for convective activity. The analysis involves the detection of contiguous cloud regions (CCRs) in the three-dimensional space of latitude, longitude, and time where Tb falls below a given threshold. A range of thresholds is considered and only CCRs that satisfy a minimum size constraint are retained in the analysis. Various statistical properties of CCRs are documented including their zonal speed of propagation, which is estimated using a Radon transformation technique. Consistent with previous studies, a majority of CCRs are found to propagate westward, typically at speeds of around 15 m s−1, regardless of underlying Tb threshold. Most of these zonally propagating CCRs have lifetimes less than 2 days and zonal widths less than 800 km, implying aggregation of just a few individual mesoscale convective systems. This object-based perspective is somewhat different than that obtained in previous Fourier-based analyses, which primarily emphasize the organization of convection on synoptic and planetary scales via wave–convection coupling. To reconcile these contrasting views, an object-based data reconstruction is developed that objectively demonstrates how the spectral peaks of synoptic- to planetary-scale waves can be attributed to the organization of CCRs into larger-scale wave envelopes. A novel method based on the randomization of CCRs in physical space leads to an empirical background spectrum for organized tropical convection that does not rely on any smoothing in spectral space. Normalization by this background reveals spectral peaks associated with synoptic- and planetary-scale waves that are consistent with previous studies.

Closed Pattern Mining from n-ary Relations

In this paper, we address the problem of closed pattern mining from n-ary relations. We propose CnS-Miner algorithm which enumerates all the closed patterns of the given n-dimensional dataset in depth first manner satisfying the user specified minimum size constraints. From the given input, the CnS-Miner algorithm generates an n-ary tree and visits the tree in depth first manner. We have proposed a generalized duplicate pruning method which prunes the subtrees that generate duplicate patterns. The space complexity of our algorithm is O(D+d) where D is the n-ary dataset and d is the depth of the tree. We have experimentally compared the proposed algorithm with DataPeeler, a recently proposed algorithm for closed pattern mining from nary relations.

Nonconvex Gate Delay Modeling and Delay Optimization

Convex delay models like the Elmore model, the related Logical Effort model, posynomial, and generalized posynomial models have always been favored by researchers, as convexity has a priori guarantees of global optimum solutions. The accuracy of the model may be sacrificed in this quest to generate convex delay models. In this paper, we investigate the use of signomial delay modeling for area/delay optimization. We present a procedure to automatically generate signomial gate delay models by nonlinear least squares fitting. As opposed to posynomial models, signomial models achieve better fits to SPICE generated data. However, signomials are not convex in general. Nevertheless, we show via duality arguments that we obtain near optimum (within 1%) solutions. Our optimization considers beta-ratio constraints, minimum and maximum size constraints for n- and p-transistors, rise/fall delays, and edge rates. The gate sizes for the fastest delay solution for a 44000-cell design, using the IBM 130-nm process, can be achieved in about 16 min of CPU time on a PC, and the area-delay tradeoff curve for 21 points can be generated in about 2 h of CPU time. To the best of our knowledge, this is the first report of using a true signomial delay model and its application to optimum gate sizing. In addition, we give performance details for the automatic data fitting for an 11-function library of static CMOS gates.