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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Eringen's model is one of the most popular theories in nonlocal elasticity. It has been applied to many practical situations with the objective of removing the anomalous stress concentrations around geometric shape singularities, which appear when the local modelling is used. Despite the great popularity of Eringen's model in mechanical engineering community, even the most basic questions such as the existence and uniqueness of solutions have been rarely considered in the research literature for this model. In this work we focus on precisely these questions, proving that the model is in general ill-posed in the case of smooth kernels, the case which appears rather often in numerical studies. We also consider the case of singular, non-smooth kernels, and for the paradigmatic case of the Riesz potential we establish the well-posedness of the model in fractional Sobolev spaces. For such a kernel, in dimension one the model reduces to the well-known fractional Laplacian. Finally, we discuss possible extensions of Eringen's model to spatially heterogeneous material distributions.

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