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

Direct correction of non-space-clamped currents via Cole's theorem

We show that one may construct a Lyapunov function for any classically damped linear system. The explicit nature of the construction permits us to show that it remains a Lyapunov function under both perturbation of the linear part and introduction of a nonlinear term. We apply our findings to a stability analysis of the discrete, as well as continuous, damped mechanical transmission line.

A Lyapunov Function for Systems Whose Linear Part is Almost Classically Damped

Our summary of the work of Keller andNiord son ( J. Math. Mech. 16, pp. 433-446, 1966) was inaccurate. We offer a correction.

http://campus.murraystate.edu/academic/faculty/mmccarthy/papers/coxmccarthy2.pdf

The Shape of the Tallest Column: Corrected

Many sensory neurons are spontaneously active when recorded in vivo or in vitro , that is, they generate action potentials at random intervals in time. For sensory neurons, spontaneous activity is usually defined as that activity recorded in the absence of any experimental sensory stimulation. Spontaneous activity is in part due to the random release of neurotransmitter at synapses, but other mechanisms presumably also contribute. In sensory receptor cells, the random activation of the transduction machinery such as isomerization of light-sensitive pigments in photoreceptors will cause the activation of downstream neurons. Similarly, ion channels in the membrane of nerve cells spontaneously open and close and thus probably also contribute to fluctuations in membrane potential that could sometimes elicit spontaneous action potentials. In this chapter, we first introduce the methods used to describe spike train variability of nerve cells and present examples of spontaneous activity patterns encountered in neurons. As we will see, these methods also help describe the variability of neuronal spike trains in responses to external stimuli.

Steven J. Cox

Papers

Direct correction of non-space-clamped currents via Cole's theorem

A Lyapunov Function for Systems Whose Linear Part is Almost Classically Damped

The Shape of the Tallest Column: Corrected

Quantification of Spike Train Variability

CHAPTER 12 – Synaptic Transmission and Quantal Release