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.

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Abstract. 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 analysisoriented methods. Their performance is illustrated through a few numerical examples.

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a b s t r a c tWe consider a model problem of isogeometric shape optimization of vibrating membranes whose shapes are allowed to vary freely. The main obstacle we face is the need for robust and inexpensive extension of a B-spline parametrization from the boundary of a domain onto its interior, a task which has to be performed in every optimization iteration. We experiment with two numerical methods (one is based on the idea of constructing a quasi-conformal mapping, whereas the other is based on a spring-based mesh model) for carrying out this task, which turn out to work sufficiently well in the present situation. We perform a number of numerical experiments with our isogeometric shape optimization algorithm and present smooth, optimized membrane shapes. Our conclusion is that isogeometric analysis fits well with shape optimization.

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We consider the problem of optimal design of flow domains for Navier–Stokes flows in order to minimize a given performance functional. We attack the problem using topology optimization techniques, or control in coefficients, which are widely known in structural optimization of solid structures for their flexibility, generality, and yet ease of use and integration with existing FEM software. Topology optimization rapidly finds its way into other areas of optimal design, yet until recently it has not been applied to problems in fluid mechanics. The success of topology optimization methods for the minimal drag design of domains for Stokes fluids (see the study of Borrvall and Petersson [12]) has lead to attempts to use the same optimization model for designing domains for incompressible Navier–Stokes flows. We show that the optimal control problem obtained as a result of such a straightforward generalization is ill‐posed, at least if attacked by the direct method of calculus of variations. We illustrate the two key difficulties with simple numerical examples and propose changes in the optimization model that allow us to overcome these difficulties. Namely, to deal with impenetrable inner walls that may appear in the flow domain we slightly relax the incompressibility constraint as typically done in penalty methods for solving the incompressible Navier–Stokes equations. In addition, to prevent discontinuous changes in the flow due to very small impenetrable parts of the domain that may disappear, we consider so‐called filtered designs, that has become a “classic” tool in the topology optimization toolbox. Technically, however, our use of filters differs significantly from their use in the structural optimization problems in solid mechanics, owing to the very unlike design parametrizations in the two models. We rigorously establish the well‐posedness of the proposed model and then discuss related computational issues.