Abstract

Abstract

In this work, we propose iterative modal solvers to generate multiphysics finite element reduced order models. We consider the strongly coupled problems defined through differential-algebraic equations with sparse discrete operators. Piezoelectric models are common examples of such problems. The approach we propose is based on the Model Order Reduction (MOR) after Implicit Schur method [1] which is used for the Krylov subspace reduction of piezoelectric devices. While their work uses the knowledge of the loading applied to the model to generate a Krylov subspace reduction basis, we propose to build a reduction basis with a priori unknown loading by modal synthesis. The basis is built from the eigenvectors of the problem after the static condensation by Schur complement of one of the physics. Typically, the Schur complement matrix is computed explicitly and it leads to dense operators [2] which limit the problem scales that can be studied due to large memory requirements and costly computations for the eigensolver used afterward. For Krylov-based eigensolvers, the most computationally difficult step is to obtain a basis spanning the eigenspace of the problem on the considered eigenvalue range. By generalizing the MOR after Implicit Schur method, this basis can be constructed by an iterative procedure using the original sparse operators instead of the dense condensed operators. The original model may be significantly larger compared to the condensed model for typical cases. However, keeping the sparsity is a critical computational advantage for the considered problems. This method is minimally intrusive for the eigensolvers that only require the implementation of a matrix-vector product. Comparing this implicit Schur complement approach to the explicit Schur complement approach shows large computational cost reductions. It also underlines the problem scale limitations of the explicit approach even on high performance computing hardware.

Abstract

A partitioned iterative method based on hierarchical decomposition is proposed for providing numerical modeling and analysis of the piezoelectric energy harvester which is involved in coupled fluid-structure interaction, coupled electro-mechanical, and a controlling electrical circuit for piezoelectric structural applications in energy harvesting. This circuitintegrated piezoelectric structural application in energy harvesting surrounded by fluid media takes the form of natural four-way coupling of fluid flow, the structure, the electromechanical effect of the piezoelectric material, and the electrical circuit. This can be formulated exactly as a fluid-structure-piezoelectric-circuit interaction. These coupled four fields are hierarchically decomposed into the fluid-structure interaction, structure-piezoelectric interaction, and piezoelectric-circuit interaction interactions. Then these subsystems are decomposed into each field. The proposed finite element method enables to reuse of existing techniques because of its modularity. Furthermore, scalability to multiphysics and multisystem couplings is expected. There are some numerical approaches in particular monolithic coupling methods are studied which are computationally expensive and leads to an ill-conditioned coefficient matrix. Nevertheless, accurate modeling for predicting the characteristics of this four-way coupling using partitioned methods has not yet been developed. This method enables an investigation of piezoelectric structures in fluid with complex geometry, material composition, and attached electrical circuits to the harvester. A flexible piezoelectric bimorph harvester in the converging channel is analyzed to demonstrate the efficiency of the proposed method. The results indicate that the method captures the coupled effect accurately.