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

A principle that has proven fruitful in modeling neural systems is to consider the simplest model capable of predicting the experimental phenomenon under consideration. This approach allows one to capture the essential points of a particular phenomenon without obscuring the picture with unnecessary details. This is precisely the approach taken by Hodgkin and Huxley to model action potential propagation along the squid giant axon in terms of sodium and potassium conductances. We have also seen how a simplification of the Hodgkin–Huxley model to a two-variable reduced FitzHugh model allows to characterize the firing properties of the Hodgkin–Huxley system in terms of phase plane analysis (Exercise 4.6). A set of simplified models are often used as a first pass to study issues related to synaptic integration or the impact of subthreshold membrane conductances on the processing of sensory inputs by neurons. In this chapter, we first present the most elementary model usually employed to simulate neurons, called the leaky integrate and fire model. Next, we introduce a class of neurons that have the ability to fire short bursts of spikes and briefly discuss their role in information processing in the nervous system. Finally, we analyze two simplified models of bursting neurons that highlight different mechanisms of burst generation within a single neuron.

Reduced Single Neuron Models

CHAPTER 10 – Reduced Single Neuron Models

Modern neuroscience can be traced back to the work of Camillo Golgi, an Italian physician and scientist who invented, in the late 1890's, a method for staining neural tissue. As the Golgi stain only “took” to a small and well separated population of cells it permitted, for the first time, one to see the trees for the forest. The Spanish neuroanatomist Santiago Ramon y Cajal took quick advantage of the Golgi method to systematically describe the different types of neurons contained in the brain of many animal species. By demonstrating that neurons, though widely varying in shape, nonetheless share common structural components, from which function may be inferred, Ramon y Cajal can be said to have founded modern neuroscience. This chapter introduces and motivates the basic electrical properties of neurons and ionic channels, including membrane conductance, capacitance, Nernst potential and synaptic conductance.

The Passive Isopotential Cell

The brain is a prodigious consumer of energy, with the neuronal signaling cost alone estimated to equal that of a human leg running a marathon. As their demand for energy increases and decreases, neurons signal the vasculature via communication through the neurovascular unit. We begin by constructing a quantitative estimate of the glucose and oxygen required to support neuronal signaling. As these molecules are not stored in the brain, we pursue an understanding, via models, of the components of the neuro-vascular unit that regulate their delivery. The blood sent to the brain consists of red blood cells that contain hemoglobin proteins to which oxygen is bound. As the hemoglobin donates its oxygen it shifts its magnetic susceptibility which can be detected and localized in what is called the Blood Oxygen Level Dependent (BOLD) signal. We explore this shift in magnetic susceptibility and its impact on the response of neighboring tissue to an external magnetic field. We then explain how this response is detected by Nuclear Magnetic Resonance and arrive at an explicit expression for the BOLD signal arising from a random network of cylindrical vessels. We then show how these detailed findings inform a unified model of the hemodynamic response. We close with a development of the mathematical tools in Magnetic Resonance Imaging that permit the BOLD signal to be observed at a resolution of millimeter scale voxels.

Neurovascular Coupling, the BOLD Signal and MRI

The first step in processing signals from the outer world is to transduce them into membrane potential changes in receptor cells at the periphery of the nervous system. In the case of light, this process is called phototransduction . Light is absorbed in photoreceptors by specialized molecules called rhodopsins that initiate a biochemical cascade of reactions resulting in a photocurrent across the membrane of the photoreceptor that leads either to depolarization or hyperpolarization of the membrane potential (depending on the photoreceptor type) and synaptic release of neurotransmitter onto postsynaptic cells. Before describing in more detail these mechanisms, we need to describe the most salient properties of light in the natural world as these have an important impact on its processing by the visual system.

Steven J. Cox

Papers

Reduced Single Neuron Models

CHAPTER 10 – Reduced Single Neuron Models

The Passive Isopotential Cell

Neurovascular Coupling, the BOLD Signal and MRI

Natural Light Signals and Phototransduction