There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

A level set method for structural topology optimization

In topology optimization of structures, materials and mechanisms, parametrization of geometry is often performed by a grey-scale density-like interpolation function. In this paper we analyze and compare the various approaches to this concept in the light of variational bounds on effective properties of composite materials. This allows us to derive simple necessary conditions for the possible realization of grey-scale via composites, leading to a physical interpretation of all feasible designs as well as the optimal design. Thus it is shown that the so-called artificial interpolation model in many circumstances actually falls within the framework of microstructurally based models. Single material and multi-material structural design in elasticity as well as in multi-physics problems is discussed.

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Material interpolation schemes in topology optimization

Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsoe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.

Topology optimization approaches: A comparative review

In this paper we seek to summarize the current knowledge about numerical instabilities such as checkerboards, mesh-dependence and local minima occurring in applications of the topology optimization method. The checkerboard problem refers to the formation of regions of alternating solid and void elements ordered in a checkerboard-like fashion. The mesh-dependence problem refers to obtaining qualitatively different solutions for different mesh-sizes or discretizations. Local minima refers to the problem of obtaining different solutions to the same discretized problem when choosing different algorithmic parameters. We review the current knowledge on why and when these problems appear, and we list the methods with which they can be avoided and discuss their advantages and disadvantages.

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Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima

Effective properties of arrangements of strong and weak materials in a checkerboard fashion are computed. Kinematic constraints are imposed so that the displacements are consistent with typical finite element approximations. It is shown that when four-node quatrilateral elements are involved, these constraints result in a numerically induced, artificially high stiffness. This can account for the formation of checkerboard patterns in continuous layout optimization problems of compliance minimization.

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Checkerboard patterns in layout optimization

A formulation for shape optimization of elastic structures subject to multiple load cases is presented. The problem is solved using a homogenization method. When compared to the single load solution strategy, it is shown that the more general formulation can produce more stable designs while it introduces little additional complexity.

Shape optimization of structures for multiple loading conditions using a homogenization method

Rationale: Deep learning is a powerful tool that may allow for improved outcome prediction.Objectives: To determine if deep learning, specifically convolutional neural network (CNN) analysis, could detect and stage chronic obstructive pulmonary disease (COPD) and predict acute respiratory disease (ARD) events and mortality in smokers.Methods: A CNN was trained using computed tomography scans from 7,983 COPDGene participants and evaluated using 1,000 nonoverlapping COPDGene participants and 1,672 ECLIPSE participants. Logistic regression (C statistic and the Hosmer-Lemeshow test) was used to assess COPD diagnosis and ARD prediction. Cox regression (C index and the Greenwood-Nam-D’Agnostino test) was used to assess mortality.Measurements and Main Results: In COPDGene, the C statistic for the detection of COPD was 0.856. A total of 51.1% of participants in COPDGene were accurately staged and 74.95% were within one stage. In ECLIPSE, 29.4% were accurately staged and 74.6% were within one stage. In COPDGene an...

Disease Staging and Prognosis in Smokers Using Deep Learning in Chest Computed Tomography.

A methodology based on topology optimization for the design of metamaterials with negative permeability is presented The formulation is based on the design of a thin layer of copper printed on a dielectric, rectangular plate of fixed dimensions An effective media theory is used to estimate the effective permeability, obtained after solving Maxwell’s equations on a representative cell of a periodic arrangement using a full 3D finite element model The effective permeability depends on the layout of copper, and the subject of the topology optimization problem is to find layouts that result in negative (real) permeability at a prescribed frequency A SIMP-like model is invoked to represent the conductivity of regions of intermediate density A number of different filtering strategies are invoked to facilitate convergence to binary solutions Examples of designs for S-band applications are presented for illustration New metamaterial concepts are uncovered, beyond the classical split-ring inspired layouts

A topology optimization method for design of negative permeability metamaterials

An overview of the method of homogenization to find the optimum layout of a linearly elastic structure is presented. The work discussed here presents a formulation to address the simultaneous optimization of the topology, shape and size of the structure. The discussion includes optimization of plane, plate and three dimensional shell structures.

Alejandro R. Diaz

Papers

Checkerboard patterns in layout optimization

Shape optimization of structures for multiple loading conditions using a homogenization method

Disease Staging and Prognosis in Smokers Using Deep Learning in Chest Computed Tomography.

A topology optimization method for design of negative permeability metamaterials

Topology and Generalized Layout Optimization of Elastic Structures