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

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

We address the optimal design problem associated with introducing passive damping devices into a structure in such a way thatall vibration frequencies are moved as far as possible into the left halfplane. As this natural objective function lacks a bounded derivative we are forced to adopt a derivative-free direct search method. We report on our application of parallel direct search to the optimal design of both shear buildings and strings.

Designing for optimal energy absorption, III: numerical minimization of the spectral abscissa

Estimating the Time Course of Pore Expansion During the Spike Phase of Exocytotic Release in Mast Cells of the Beige Mouse

One may produce the qth harmonic of a string of length π by applying the 'correct touch' at the node π/q during a simultaneous pluck or bow. This notion was made precise by a model of Bamberger, Rauch and Taylor. Their 'touch' is a damper of magnitude b concentrated at π/q .T he 'correct touch' is that b for which the modes, that do not vanish at π/q, are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree q � 1. We establish lower and upper bounds on the spectral abscissa and show that the set of associated root vectors constitutes a Riesz basis and so identify 'correct touch' with the b that minimizes the spectral abscissa.

/pdf/eliciting-harmonics-on-strings-28wvp8srpm.pdf

Eliciting harmonics on strings

We consider the wave equation on a finite interval with fixed ends and nonuniform viscous damping. We prove there can be at most one even damping associated with a given spectrum. We then develop a refined asymptotic formula for the high frequencies. When the damping is even about the domain's midpoint, terms in this expansion are Fourier coefficients for functions of the damping. Furthermore, the expansion is often quite accurate even at low frequencies, thus suggesting a simple numerical procedure for reconstructing even damping coefficients from measured eigenvalues. Computational examples illustrate the efficacy of this procedure, even in the presence of noise.

/pdf/reconstructing-an-even-damping-from-a-single-spectrum-1k3znqrom1.pdf

Steven J. Cox

Papers

Designing for optimal energy absorption, III: numerical minimization of the spectral abscissa

Estimating the Time Course of Pore Expansion During the Spike Phase of Exocytotic Release in Mast Cells of the Beige Mouse

Eliciting harmonics on strings

Reconstructing an even damping from a single spectrum

A Fast, Fully Implicit Backward Euler Solver for Dendritic Neurons