Anton Evgrafov

Design of Piezoelectric Energy Harvesting Systems: A Topology Optimization Approach Based on Multilayer Plates and Shells

We develop a computational approach to analyze and design piezoelectric energy harvesting systems composed of layered plates and shells connected to an electrical circuit. The finite element method is used to model the coupled electromechanics of the piezoelectric harvesting structure and a lumped parameter model for the dynamics of the electrical circuit. We assume the harvester is subjected to a prescribed harmonic base excitation and that the structural and electrical responses are linear. We use topology optimization to design the layout of a multilayer structure consisting of structural, piezoelectric, and electrode layers, as well as the electrical circuit. The flexibility of our formalism admits the definition of specific system-level objectives, e.g., maximize the power harvested, in an algebraic fashion. After describing our analysis and design approaches, we present examples that demonstrate the versatility of our approach and show how it can be used to explore general behavior and develop overarching design principles for piezoelectric energy harvesting devices. For the objective of maximizing the power harvested, we investigate: (i) optimal designs for various piezoelectric to substrate thickness ratios, (ii) the effect of mass loading on optimal design, and (iii) the sensitivity of designs to shape variations.

Planar Parametrization in Isogeometric Analysis

Before isogeometric analysis can be applied to solving a partial differential equation posed over some physical domain, one needs to construct a valid parametrization of the geometry. The accuracy of the analysis is affected by the quality of the parametrization. The challenge of computing and maintaining a valid geometry parametrization is particularly relevant in applications of isogemetric analysis to shape optimization, where the geometry varies from one optimization iteration to another. We propose a general framework for handling the geometry parametrization in isogeometric analysis and shape optimization. It utilizes an expensive non-linear method for constructing/updating a high quality reference parametrization, and an inexpensive linear method for maintaining the parametrization in the vicinity of the reference one. We describe several linear and non-linear parametrization methods, which are suitable for our framework. The non-linear methods we consider are based on solving a constrained optimization problem numerically, and are divided into two classes, geometry-oriented methods and analysis-oriented methods. Their performance is illustrated through a few numerical examples.

ISOGEOMETRIC SHAPE OPTIMIZATION FOR ELECTROMAGNETIC SCATTERING PROBLEMS

We consider the benchmark problem of magnetic energy density enhancement in a small spatial region by varying the shape of two symmetric conducting scatterers. We view this problem as a prototype for a wide variety of geometric design problems in electromagnetic applications. Our approach for solving this problem is based on shape optimization and isogeometric analysis. One of the major di-culties we face to make these methods work together is the need to maintain a valid parametrization of the computational domain during the optimization. Our approach to generating a domain parametrization is based on minimizing a second order approximation to the Winslow functional in the vicinity of a reference parametrization. Furthermore, we enforce the validity of the parametrization by ensuring the non-negativity of the coe-cients of a B-spline expansion of the Jacobian. The shape found by this approach outperforms earlier design computed using topology optimization by a factor of one billion.

Topology Optimization of Fluid Problems by the Lattice Boltzmann Method

We consider the optimal design of flow domains for Navier-Stokes. The problem is solved by a topology optimization approach varying the effective porosity of a fictitious material. The boundaries of the flow domain are represented by potentially discontinuous material distributions. Navier-Stokes flows are traditionally approximated by finite element and finite volume methods. These schemes, however, are particularly sensitive to the discretization of the flow along the boundaries, leading to significant robustness issues in the case of non-smooth boundary representations. Therefore, we study the potential of the lattice Boltzmann method for approximating low Mach-number incompressible viscous flows for topology optimization. In the lattice Boltzmann method the geometry of flow domains is defined in a discontinuous manner, similar to the approach used in material based topology optimization. In addition, this nontraditional discretization method features parallel scalability and allows for high-resolution fluid meshes. In this paper, we show how the variation of the porosity can be used in conjunction with the lattice Boltzmann method for the optimal design of fluid domains. An adjoint formulation of the sensitivity equations will be presented and the potential of this topology optimization approach will be illustrated by a numerical examples.

A parallel Schur complement solver for the solution of the adjoint steady-state lattice Boltzmann equations: application to design optimisation

We study the numerical performance of a parallel Schur complement method towards the solution of the steady-state lattice Boltzmann equations on large meshes as needed for parametric design optimisation problems, in particular topology optimisation. For the gradient-based topology optimisation framework a porosity model is used to continuously transition from fluid to solid. Deriving the sensitivity equations for the lattice Boltzmann method (LBM), we identify the Jacobian of the LBM fixed-point formulation, which poses the key computational challenge in LBM topology optimisation due to its large size. We show that the Schur complement method can be used to decompose the fixed-point Jacobian, making the solution of the LBM sensitivity equations more memory efficient, in particular for 2D problems. The effectiveness of the overall lattice Boltzmann based topology optimisation framework is illustrated with the optimal design of a micro valve, posted by a dual-objective optimisation problem.

Iso-geometric shape optimization of magnetic density separators

Purpose – The waste recycling industry increasingly relies on magnetic density separators. These devices generate an upward magnetic force in ferro-fluids allowing to separate the immersed particles according to their mass density. Recently, a new separator design has been proposed that significantly reduces the required amount of permanent magnet material. The purpose of this paper is to alleviate the undesired end-effects in this design by altering the shape of the ferromagnetic covers of the individual poles. Design/methodology/approach – The paper represents the shape of the ferromagnetic pole covers with B-splines and defines a cost functional that measures the non-uniformity of the magnetic field in an area above the poles. The authors apply an iso-geometric shape optimization procedure, which allows us to accurately represent, analyze and optimize the geometry using only a few design variables. The design problem is regularized by imposing constraints that enforce the convexity of the pole cover sh...

Hydrodynamics studies of cyclic voltammetry for electrochemical micro biosensors

We investigate the effect of flow rate on the electrical current response to the applied voltage in a micro electrochemical system. To accomplish this, we considered an ion-transport model that is governed by the Nernst-Planck equation coupled to the Navier-Stokes equations for hydrodynamics. The Butler-Volmer relation provides the boundary conditions, which represent reaction kinetics at the electrode-electrolyte interface. The result shows that convection drastically affects the rate of surface kinetics. At a physically sufficient high flow rates and lower scan rates, the current response is limited by the convection due to fresh ions being brought to the electrode surface and immediately taken away before any surface reaction. However, at high flow and scan rates, the Faradaic current overrides current due to convection. The model also allows predicting the effect of varying electrolyte concentration and scan rates respectively.

Design of nanostructured phononic materials

The ability of a material containing a periodic arrangement of second-phase inclusions to prevent transmission of waves in certain frequency ranges is well known. This is true for all types of waves including acoustic, electromagnetic, and elastic. These forbidden regions are called band gaps. They arise as incident waves are effectively attenuated by interference among the scattered wave fields. Indeed much of current semiconductor technology revolves around band-gap engineering with regard to electron flow in the periodic potentials resulting from atoms in their lattice positions. The phenomena are also being heavily explored in the context of light via the development of photonic crystals. Things become more interesting if instead of thinking of periodic arrangements, one selectively removes some of the inclusions in the periodic geometry creating defects. If done right, this can result in a material microstructure that can guide waves through the material. Advances in nano and micromanufacturing technologies in the last couple of years have opened up the possibility to fabricate heterogeneous material systems with precise positional control of the constituent materials. For example, it is now possible to place thin-film materials precisely at a resolution of fractions of a micron. Depending on how it is done, one can envision designing a material so that a wave will be guided to a particular location and/or away from another and as a result damping or amplifying the wave locally. In this work we develop a topology optimization approach to design such nanostructured materials. We demonstrate the approach through the design of three multifunctional phononic composite materials composed of silicon and aluminum: i) a grating designed to stop wave propagation at a specified frequency, ii) a waveguide that bends the propagation path of an elastic wave, and iii) an elastic switch that switches an input signal between two output ports based on the state of the input signal.

Discontinuous Petrov-Galerkin Methods for Topology Optimization

Discontinuous Petrov–Galerkin (DPG) methods constitute a modern class of finite element methods, which present several advantages when compared with traditional Bubnov–Galerkin methods, especially when the latter is applied to indefinite or non-symmetric problems. Our objective is to utilize the advantages of DPG methods in the context of topology optimization.

Hot Blade Cuttings for the Building Industries

The constructions of advanced architectural designs are presently very labour intensive, time consuming, and expensive. They are therefore only applied to a few prestige projects, and it is a major challenge for the building industry to bring the costs down and thereby offer the architects more variability in the (economically allowed) designs—i.e., to allow them to think out of the box. To address this challenge The Danish National Advanced Technology Foundation (now Innovation Fund Denmark) is currently supporting the BladeRunner project that involves several Danish companies and public institutions. The project aims to reduce the amount of manual labour as well as production time by applying robots to cut expanded polystyrene (EPS) moulds for the concrete to form doubly curved surfaces. The scheme is based upon the so-called Hot Wire or Hot Blade technology where the surfaces are essentially swept out by driving an Euler elastica through a block of EPS. This paper will be centered around the mathematical challenges encountered in the implementation of this idea. Since the elastica themselves are well known and described in the works of Euler et al. already in eighteenth century, these new challenges are mainly concerned with the rationalization of the architects’ CAD drawings into surfaces that can be created via this particular sweeping and cutting technology.