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.

/pdf/material-interpolation-schemes-in-topology-optimization-1ps0tr57ei.pdf

Material interpolation schemes in topology optimization

A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.

/pdf/nonlinear-model-reduction-via-discrete-empirical-f3tjqvqhuf.pdf

Nonlinear Model Reduction via Discrete Empirical Interpolation

Recent advancements in computational inverse-design approaches — algorithmic techniques for discovering optical structures based on desired functional characteristics — have begun to reshape the landscape of structures available to nanophotonics. Here, we outline a cross-section of key developments in this emerging field of photonic optimization: moving from a recap of foundational results to motivation of applications in nonlinear, topological, near-field and on-chip optics. Starting with a desired optical output it is possible to use computational algorithms to inverse design devices. The approach is reviewed here with an emphasis on nanophotonics.

Inverse design in nanophotonics

Phononic band-gap materials prevent elastic waves in certain frequency ranges from propagating, and they may therefore be used to generate frequency filters, as beam splitters, as sound or vibration protection devices, or as waveguides. In this work we show how topology optimization can be used to design and optimize periodic materials and structures exhibiting phononic band gaps. Firstly, we optimize infinitely periodic band-gap materials by maximizing the relative size of the band gaps. Then, finite structures subjected to periodic loading are optimized in order to either minimize the structural response along boundaries (wave damping) or maximize the response at certain boundary locations (waveguiding).

Systematic design of phononic band-gap materials and structures by topology optimization.

Data-driven discovery is revolutionizing the modeling, prediction, and control of complex systems. This textbook brings together machine learning, engineering mathematics, and mathematical physics to integrate modeling and control of dynamical systems with modern methods in data science. It highlights many of the recent advances in scientific computing that enable data-driven methods to be applied to a diverse range of complex systems, such as turbulence, the brain, climate, epidemiology, finance, robotics, and autonomy. Aimed at advanced undergraduate and beginning graduate students in the engineering and physical sciences, the text presents a range of topics and methods from introductory to state of the art.

Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control

The Singular Value Decomposition (SVD) is a natural matrix factorization that offers one a quantitative means of discerning what is, and what is not, of importance in the underlying data or model. We build this factorization from ingredients we assembled in our study of the eigen-decomposition of symmetric matrices in Chapter 6. We apply this factorization, in its guise as Principal Component Analysis (PCA), to data reduction in the context of sorting spikes that reach a single recording electrode from multiple sources. PCA is a technique for choosing coordinates in which the data exhibits maximal variance. We observe that a simple Hebbian learning rule achieves the same outcome. We then demonstrate how the SVD may be used to reduce the dimension of dynamical models. We show that a 400-dimensional quasi-active cable may be accurately simulated with as few as five variables.

The Singular Value Decomposition and Applications

We consider neurons with large dendritic trees that are weakly excitable in the sense that back propagating action potentials are severly attenuated as they travel from the small, strongly excitable, spike initiation zone. In previous work we have shown that the computational size of weakly excitable cell models may be reduced by two or more orders of magnitude, and that the size of strongly excitable models may be reduced by at least one order of magnitude, without sacrificing the spatio-temporal nature of its inputs (in the sense we reproduce the cell's precise mapping of inputs to outputs). We combine the best of these two strategies via a predictor--corrector decomposition scheme and achieve a drastically reduced highly accurate model of a caricature of the neuron responsible for collision detection in the locust.

/pdf/model-reduction-of-strong-weak-neurons-55l6h4db2g.pdf

Model reduction of strong-weak neurons.

We study the average temperature in a homogeneous disk subject to uniform heating in its interior and Newton's law of cooling, 
$hv+\partial v/\partial n=0$
 , on its boundary. More precisely, among those h taking values in a prescribed interval, and of prescribed mean, we identify the minimizer and a maximizer of the average temperature. The latter characterization makes use of circular symmetrization. We include an elementary proof of the fact that such symmetrization does not increase the Dirichlet energy of any H

 1 function on the disk.

Circular symmetrization and extremal Robin conditions

Signal detection theory, as its name implies, is the mathematical theory used to optimally detect signals embedded in noise. If the noise is a random variable with a known probability distribution, then it is possible to exploit this knowledge to determine an optimal method of detecting the signal. More generally, we may think of signal detection in noise as the testing of an hypothesis with two alternatives: is the signal present or absent in the noisy background? Under this formulation the theory is broadly applicable to neuroscience to quantify the information conveyed by neurons or ensemble of neurons about sensory stimuli. In §§27.1–27.3 we introduce the basic concepts and results of signal detection theory. In §§27.4 and 27.5 we generalize them to multi-dimensional signals, a step that allows us to analyze the coding of information in population of neurons and in the time-varying firing rate or membrane potential of single neurons.

Signal Detection Theory

Permitted to fasten an elastic body on but half of its boundary, which portion should be fastened in order to minimize the work done by a specified load? For both plates and planar sheets we establish the existence of a unique solution to a relaxed version of the problem and present the results of a detailed numerical investigation.

/pdf/minimal-compliance-fastening-of-elastic-bodies-21pv4az6yl.pdf

Steven J. Cox

Papers

The Singular Value Decomposition and Applications

Model reduction of strong-weak neurons.

Circular symmetrization and extremal Robin conditions

Signal Detection Theory

Minimal compliance fastening of elastic bodies