## Final Report Summary - FSI-HARVEST (Numerical modelling of smart energy harvesting devicesdriven by flow-induced vibrations)

The project investigates a new class of piezo-electric energy harvesting devices for renewable energy resources. The key idea is to invert the traditional intention of engineers to avoid flow-induced excitation of structures such, that fluid-structure interaction can successfully be controlled and utilised in order to provide independent power supply to small-scale electrical devices. Possible application are e.g. micro electro-mechanical systems, monitoring sensors at remote locations or even in-vivo medical devices with the advantage of increased independence on local energy storage and reduced maintenance effort. This energy converter technology involves transient boundary-coupled fluid-structure interaction, volume-coupled piezo-electric-mechanics as well as a controlling electric circuit simultaneously. In order to understand the phenomenology and to increase robustness and performance of such devices, a mathematical and numerical model of the transient strongly-coupled non-linear multi-physics system is to be developed in the first project period and utilised for systematic computational analyses in the second half of the research project.

In order to meet the primary objective of the first project period -- the development of a monolithic approach to the numerical simulation of FSI energy harvesting devices including the flow-structure problem, piezo-ceramic material and electric circuit -- a consistent and reliable mathematical formulation of the coupled multi-physics system of the energy harvester (B3.2.1 of work plan: milestone A - Theoretical background - 6 months) was established and the deliverable of a "profound mathematical formulation of the coupled multi-field problem was realised. The properties of the resulting set of equations requires efficient methods for obtaining approximate solutions (B3.2.1 of work plan: milestone B - Discretisation - 18 months). The method of weighted residuals is utilised to arrive at the weak form of the governing equations of the multi-physics problem in 3D. The resulting integral equations of fluid, oscillator, piezoelectric patches and the attached electric circuit are uniformly discretised both in space and time using the space-time finite element method in a mixed-hybrid scheme. Time integration is performed with the discontinuous Galerkin method. The non-linear equations are linearised and solved using a Newton-Raphson scheme. Taking advantage of a native coupling of fluid and structure through the velocity-based formulation, a simultaneous solution strategy is obtained such that the strongly coupled system is described as a single algebraic system and the unknowns are solved for simultaneously. From exchange with other researchers in the field it is deduced that the holistic approach followed in this work is required to successfully capture the coupled phenomenology of the involved fields including the energy status accurately. The space-finite-element-based computational framework has been largely implemented into an efficient in-house computer code based on high-performance computing libraries. Verification and validation tests confirm the accuracy of the numerical solution software generated for the individual physical fields and the multi-physical setup. The deliverable of a reliable space-time finite element model for piezo-electric material and electrical circuit was realised.

The second half of the project is dedicated to the verification and validation of the predictive model for piezoelectric energy harvesting devices developed during the first project period (B3.2.1 of work plan: milestone C - Validation and phenomenological parameter study on full model). In a first step, the computational framework was applied to a 'dry configuration', involving the harvester structure and the electrical circuit. The source of mechanical vibration is provided by an harmonic base excitation of the clamped bimorph plate. This configuration was selected for the initial validation since analytical and experimental results are available for this kind of system from a publication by Ertuk and Inman. It could be shown that the time-domain computational model (using a space-time finite element formulation) developed in the first phase of this CIG is able to predict the results obtained experimentally by Ertuk and Inman very closely, both in the transient as well as the quasi-steady regime. The deliverable "verification and validation of the fully coupled model" was realised. Additionally, a frequency-domain analysis variant of a linearised version of the coupled predictive model was developed and tested. The frequency-domain variant allows for an holistic analysis of the fully coupled electromechanical-circuit system and by this can predict with just one computation those operation states that are optimal for a given harvester setup with respect to the power output generated. In this form, the holistic frequency-domain model is unique and the first of its kind in the area of energy harvesting devices.

With a number of numerical studies, the influence of certain system properties on the generated power output could be described parametrically. The deliverable "identification of optimal operation states" was therefore realised. Having both, time and frequency domain analysis, reliably available allowed to proceed in a second step with the analysis of the fluid-structure coupled system setup 'wet configuration'). Here, the mechanical excitation of the system is provided in terms of fluid pressure fluctuations on the structure. The cantilever plate setup used above was placed in a surrounding laminar flow field with a bluff obstacle at the leading edge and subjected to vortex induced vibrations. Since the response of the structural harvester system is then in the first bending mode, a classical bimorph setup with electrodes covering the full top and bottom of the piezoelectric layers can be used. A more interesting situation was identified for the more effective scenario of movement-induced vibrations of the plate (flutter), since then the structural response is in the second bending mode. A classical bimorph setup would lead to strain cancellations and therefore to much reduced power output. With the 3D finite element framework developed in this CIG it is easily possible to investigate more effective alternative configurations using interdigitated electrodes on more complex geometries or gradually varying material properties.

In order to meet the primary objective of the first project period -- the development of a monolithic approach to the numerical simulation of FSI energy harvesting devices including the flow-structure problem, piezo-ceramic material and electric circuit -- a consistent and reliable mathematical formulation of the coupled multi-physics system of the energy harvester (B3.2.1 of work plan: milestone A - Theoretical background - 6 months) was established and the deliverable of a "profound mathematical formulation of the coupled multi-field problem was realised. The properties of the resulting set of equations requires efficient methods for obtaining approximate solutions (B3.2.1 of work plan: milestone B - Discretisation - 18 months). The method of weighted residuals is utilised to arrive at the weak form of the governing equations of the multi-physics problem in 3D. The resulting integral equations of fluid, oscillator, piezoelectric patches and the attached electric circuit are uniformly discretised both in space and time using the space-time finite element method in a mixed-hybrid scheme. Time integration is performed with the discontinuous Galerkin method. The non-linear equations are linearised and solved using a Newton-Raphson scheme. Taking advantage of a native coupling of fluid and structure through the velocity-based formulation, a simultaneous solution strategy is obtained such that the strongly coupled system is described as a single algebraic system and the unknowns are solved for simultaneously. From exchange with other researchers in the field it is deduced that the holistic approach followed in this work is required to successfully capture the coupled phenomenology of the involved fields including the energy status accurately. The space-finite-element-based computational framework has been largely implemented into an efficient in-house computer code based on high-performance computing libraries. Verification and validation tests confirm the accuracy of the numerical solution software generated for the individual physical fields and the multi-physical setup. The deliverable of a reliable space-time finite element model for piezo-electric material and electrical circuit was realised.

The second half of the project is dedicated to the verification and validation of the predictive model for piezoelectric energy harvesting devices developed during the first project period (B3.2.1 of work plan: milestone C - Validation and phenomenological parameter study on full model). In a first step, the computational framework was applied to a 'dry configuration', involving the harvester structure and the electrical circuit. The source of mechanical vibration is provided by an harmonic base excitation of the clamped bimorph plate. This configuration was selected for the initial validation since analytical and experimental results are available for this kind of system from a publication by Ertuk and Inman. It could be shown that the time-domain computational model (using a space-time finite element formulation) developed in the first phase of this CIG is able to predict the results obtained experimentally by Ertuk and Inman very closely, both in the transient as well as the quasi-steady regime. The deliverable "verification and validation of the fully coupled model" was realised. Additionally, a frequency-domain analysis variant of a linearised version of the coupled predictive model was developed and tested. The frequency-domain variant allows for an holistic analysis of the fully coupled electromechanical-circuit system and by this can predict with just one computation those operation states that are optimal for a given harvester setup with respect to the power output generated. In this form, the holistic frequency-domain model is unique and the first of its kind in the area of energy harvesting devices.

With a number of numerical studies, the influence of certain system properties on the generated power output could be described parametrically. The deliverable "identification of optimal operation states" was therefore realised. Having both, time and frequency domain analysis, reliably available allowed to proceed in a second step with the analysis of the fluid-structure coupled system setup 'wet configuration'). Here, the mechanical excitation of the system is provided in terms of fluid pressure fluctuations on the structure. The cantilever plate setup used above was placed in a surrounding laminar flow field with a bluff obstacle at the leading edge and subjected to vortex induced vibrations. Since the response of the structural harvester system is then in the first bending mode, a classical bimorph setup with electrodes covering the full top and bottom of the piezoelectric layers can be used. A more interesting situation was identified for the more effective scenario of movement-induced vibrations of the plate (flutter), since then the structural response is in the second bending mode. A classical bimorph setup would lead to strain cancellations and therefore to much reduced power output. With the 3D finite element framework developed in this CIG it is easily possible to investigate more effective alternative configurations using interdigitated electrodes on more complex geometries or gradually varying material properties.