Complicated Geometry Research Articles

Evaluation of the Inter-Building Effect on Energy Saving Potential of Radiative Cooling Materials

Buildings in the urban area have huge decarbonization potential by applying novel energy-saving technologies. Recently, radiative cooling materials have drawn much attention for their refrigerant-free and energy-free properties in reducing energy consumption. However, the complicated geometry and configuration of artificial structures in urban areas can significantly limit the actual application of radiative material. One limitation is the inter-building effect with shading on buildings. Therefore, this study developed a fast solar irradiation generator on buildings' envelope, and radiative materials' energy-saving potential was evaluated on buildings' wall areas with different irradiation strengths. A case study showed that radiative materials have higher cooling performance on the wall area with solar irradiation larger than 30 Wh/m2.

Crack Initiation in Compacted Graphite Iron with Random Microstructure: Effect of Volume Fraction and Distribution of Particles.

Thanks to the distinctive morphology of graphite particles in its microstructure, compacted graphite iron (CGI) exhibits excellent thermal conductivity together with high strength and durability. CGI is extensively used in many applications, e.g., engine cylinder heads and brakes. The structural integrity of such metal-matrix materials is controlled by the generation and growth of microcracks. Although the effects of the volume fraction and morphology of graphite inclusions on the tensile response of CGI were investigated in recent years, their influence on crack initiation is still unknown. Experimental studies of crack initiation require a considerable amount of time and resources due to the highly complicated geometries of graphite inclusions scattered throughout the metallic matrix. Therefore, developing a 2D computational framework for CGI with a random microstructure capable of predicting the crack initiation and path is desirable. In this work, an integrated numerical model is developed for the analysis of the effects of volume fraction and nodularity on the mechanical properties of CGI as well as its damage and failure behaviours. Finite-element models of random microstructure are generated using an in-house Python script. The determination of spacings between a graphite inclusion and its four adjacent particles is performed with a plugin, written in Java and implemented in ImageJ. To analyse the orientation effect of inclusions, a statistical analysis is implemented for representative elements in this research. Further, Johnson-Cook damage criteria are used to predict crack initiation in the developed models. The numerical simulations are validated with conventional tensile-test data. The created models can support the understanding of the fracture behaviour of CGI under mechanical load, and the proposed approach can be utilised to design metal-matrix composites with optimised mechanical properties and performance.

Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport

We introduce an hp-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and hp-version a priori error analysis of the proposed method, by deriving suitable hp-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.

Efficient part orientation algorithm for additive manufacturing in industrial applications

Over the past few decades, the scientific community’s and industry’s interest in additive manufacturing technologies has surged. This technology is distinguished by the layer-by-layer deposition of the raw materials and the piece’s growth in a predetermined build orientation. This factor impacts the process’ overall cost, surface quality, and other crucial parameters. Numerous methods to solve competing aspects have been proposed in the literature, with the more promising that iteratively uses ray-tracing techniques. Existing algorithms iterate for each discrete element of the model’s bounding box projection onto the building platform. However, when optimisation algorithms are used on real-life industrial parts, computational time problems arise due to the high number of faces in the models. A new computational technique to determine the appropriate part orientation to reduce the support volume is proposed to address the problem. The method reduces the computational time, cycling the ray-tracing only on the triangles where the model surface is discretised. This approach has been integrated into an enhanced particle swarm optimisation algorithm to prove its efficiency. The approach is intended for industrial applications where it is necessary to handle complicated geometries quickly and efficiently to find the best orientation. Based on the computer’s resources and the complexity of the faceted model, a set of case studies with an industrial engineering significance is used to demonstrate the approach’s effectiveness.

Parameter automatic optimization strategy for laser powder bed fusion using neural network infrared radiation intensity prediction model

Conventional scanning approaches with fixed laser parameters are prone to causing localized defects and in-homogeneous performance when forming parts with complicated geometries using the laser powder bed fusion (LPBF) process. While online monitoring of the forming process using thermal cameras allows the identification of abnormal infrared radiation intensity values of the scanning line to pinpoint regions of potential defects, it fails to proactively prevent defects from occurring. To solve this problem, this study constructed a back propagation (BP) neural network to predict the infrared radiation intensity of the current layer before scanning. In the meantime, this study designed a parameter optimization algorithm to adjust the local scanning parameters with a view to avoiding defects and thus enhancing the performance consistency of the formed parts. This study drew on the idea of the divide-and-conquer method to divide the parts into multiple units for data acquisition by thermal camera, and has collected 500,000 pieces of experimental data through 15 experiments. By utilizing these data, this study trained a BP neural network to predict the infrared radiation intensity, and its mean absolute error (MAE) of the prediction results was 25, the coefficient of determination (R2) was 0.708, and the F1 score (abnormal radiation intensity identification and classification) was 0.802. Subsequently, the study optimized the scanning parameters of the unit by using particle swarm optimization (PSO), and performed practical application and validation on spherical gear parts of 316 L stainless steel and blade parts of DZ125 nickel-based superalloy. Findings indicated that the automatic optimization strategy can effectively prevent abnormal infrared radiation intensity values during the scanning process and enhance the performance consistency of the LPBF process for complicated parts in both different materials and structures.

On the dynamics of the elastic concrete systems reinforced by advanced nanocomposites

This work investigates the vibrational properties of doubly curved elastic concrete panels reinforced with multi-hybrid nanocomposites under in-plane loading. These cutting-edge materials use the advantages of several nanocomposites to produce greater performance by integrating many kinds of nanofibers and nanotubes to boost mechanical qualities. The study focuses on the effects of these reinforcements on the dynamic behavior of doubly curved panels, which are a typical structural component in contemporary engineering applications including civil infrastructure, automotive, and aerospace. The study investigates the natural frequencies and dynamic stability of the panels under in-plane stress circumstances using a thorough analytical method. The complicated geometry and material heterogeneity are modeled using mathematical modeling, which offers comprehensive insights into the impact of nanocomposite reinforcements. The findings show that, in comparison to conventional concrete panels, multi-hybrid nanocomposites considerably increase stiffness characteristics and vibrational performance. The results highlight the potential of these novel materials to transform structural component design and application, resulting in more robust and effective engineering solutions. By improving our knowledge of hybrid nanocomposite behavior under dynamic stresses, our study opens the door to a wider use of these materials science and engineering principles in materials science and engineering.

A novel titanium alloy for load-bearing biomedical implants: Evaluating the antibacterial and biocompatibility of Ti536 produced via electron beam powder bed fusion additive manufacturing process

Additive manufacturing (AM) of Ti-based biomedical implants is a pivotal research topic because of its ability to produce implants with complicated geometries. Despite desirable mechanical properties and biocompatibility of Ti alloys, one major drawback is their lack of inherent antibacterial properties, increasing the risk of postoperative infections. Hence, this research focuses on the Ti536 (Ti5Al3V6Cu) alloy, developed through Electron Beam Powder Bed Fusion (EB-PBF), exploring bio-corrosion, antibacterial features, and cell biocompatibility. The microstructural characterization revealed grain refinement and the formation of Ti2Cu precipitates with different morphologies and sizes in the Ti matrix. Electrochemical tests showed that Cu content minimally influenced the corrosion current density, while it slightly affected the stability, defect density, and chemical composition of the passive film. According to the findings, the Ti536 alloy demonstrated enhanced antibacterial properties without compromising its cell biocompatibility and corrosion behavior, thanks to Ti2Cu precipitates. This can be attributed to both the release of Cu ions and the Ti2Cu precipitates. The current study suggests that the EB-PBF fabricated Ti536 sample is well-suited for use in load-bearing applications within the medical industry. This research also offers an alloy design roadmap for novel biomedical Ti-based alloys with superior biological performance using AM methods.

Performance Assessment Analyzes of Research Nuclear Reactors Based on Steady-state and Kinetic Models

The mainstays of nuclear substance radiation and isotopic synthesis are nuclear-powered power plants, however effective safety evaluation is made tougher by the complicated construction topologies and physical connection effects. This work proposes a multiphysics-linked technique for evaluating both the kinetic and steady-state behaviors of the MPRR and LVR-15 laboratory reactors. To represent complicated member geometries, homogenized assembling sections are generated using two-dimensional whole-core computational simulations. It is discovered that the steady-state findings and the so-called Monte Carl solution comparisons correspond quite nicely. The greatest assemble power mistakes for LVR-15 and MPRR are 6.49%/10%, and the highest command rod value mistakes are 31 pcm/136 pcm, and the mistakes are 377 pcm/383 pcm, accordingly. Meanwhile, the study is done on transitory procedures, such as reactivity-initiated disasters and exposed loss-of-flow mishaps. Both units' modeling findings show plausible adverse feedback events. Furthermore, it is shown that the two reactors' accident-related behaviors are comparable though having different core architectures since they employ the exact same kinds of water as a fluid. The technique for studying nuclear power plant kinetics known as Multi-Physics Simulation (MPM) is explained. Drawing on many research and verification efforts conducted at Politecnico di Milan, Italy, MPM is shown to be a valuable instrument for managing reactors' security and oversight. It may be viewed as a holistic analytical tool that is implemented during the reactor architecture design phase. The capacity to concurrently answer the interrelated equations that control the many physical processes taking place in a nuclear plant inside the same simulated setting is a core characteristic of MPM.

Large-eddy simulation of wall-bounded incompressible turbulent flows based on multi-moment finite volume formulation

The multi-moment based numerical model is proposed for performing large-eddy simulation (LES) of wall-bounded incompressible turbulent flows on the unstructured grids. In this model, velocity is defined as volume integral average (VIA) and point values (PVs) which are updated as prognostic variables simultaneously in time. Using VIA and PV in each cell, this method constructs higher-order polynomial on a compact stencil and more importantly, provides a new approach to design subgrid-scale (SGS) models for LES. Various SGS models are developed in this framework where the eddy-viscosity is computed separately at both VIA and PVs using the least-square method on different interpolation stencils. Compared with conventional finite volume schemes, the proposed method effectively reduces numerical dissipation and largely suppresses the sensitivity of numerical solution on SGS models and resolves more elaborated flow structures on the same mesh resolution. It also reduces solution dependence on the shape and quality of grid elements, as demonstrated by comparing numerical results on the different types of grids. The proposed method can attain high-fidelity simulation without loss of numerical efficiency and robustness, which is highly appealing for complex turbulent problems with high Reynolds numbers and complicated geometries.

Microwave Welding of Thermoplastic using Silicon Carbide Nanowhiskers as Susceptor: Effect of Heating Duration

Microwave welding is becoming more popular than conventional joining methods due to its advantages such as rapid and localised heating as well as applicable to components with complicated geometry. Previously reported susceptor, such as carbonaceous materials and conductive polymers, are toxic and the welding process involving these susceptors is time-consuming. Because of its exceptional microwave absorption and biocompatibility, silicon carbide nanowhiskers (SiCNWs) was employed as the microwave susceptor for microwave welding. Microwave welding in this study comprises of only three simple steps: SiCNWs suspension preparation, SiCNWs application and microwave heating. The weld strength of welded joint was then characterised using tensile test and energy dispersive x-ray spectroscopy equipped scanning electron microscopy (EDS-SEM) to study its mechanical properties and cross-section microstructure. The influence of microwave irradiation time was studied in this study, and it is found that the weld strength rose with the extension of microwave irradiation time, until a maximum weld strength of 1.61 MPa was achieved by 17 s welded joint. The development of SiCNWs reinforced PP nanocomposite welded joint layer is responsible for the enhanced weld strength. Prolonged heating duration may also result in flaws such as void formation at the welded joint, which subsequently lowered the weld strength to 0.60 MPa when the heating duration was extended to 20 s. In sum, a strengthen welded joint can be formed with rapid microwave heating under the proper control of heating duration.

KIC 4150611: A quadruply eclipsing heptuple star system with a g-mode period-spacing pattern. Eclipse modelling of the triple and spectroscopic analysis.

KIC 4150611 is a high-order multiple composed of a triple system.\ It comprises: (1) a F1V primary (Aa) that is eclipsed on a 94.2d period by a tight 1.52d binary composed of two dim K/M dwarfs (Ab1 and Ab2), which also eclipse each other; (2) an 8.65d eccentric, eclipsing binary composed of two G stars (Ba and Bb); and (3) another faint eclipsing binary composed of two stars of unknown spectral type (Ca and Cb). In addition to its many eclipses, the system is an SB3 spectroscopic multiple (Aa, Ba, and Bb), and the primary (Aa) is a hybrid pulsator that exhibits high amplitude pressure and gravity modes. In aggregate, this richness in physics offers an excellent opportunity to obtain a precise physical characterisation of some of the stars in this system. In this work we aim to characterise the F1V primary by modelling its complex eclipse geometry and disentangled stellar spectra in preparation for follow-up work that will focus on its pulsations. We employed a novel photometric analysis of the complicated eclipse geometry of Aa to obtain the orbital and stellar properties of the triple. We acquired 51 TRES spectra at the Fred L. Whipple Observatory, calculating radial velocities and orbital elements of Aa (SB1) and the B binary (SB2). These spectra and radial velocities were used to perform spectral disentangling for Aa, Ba, and Bb. Spectral modelling was applied to the disentangled spectrum of Aa to obtain atmospheric properties. From our eclipse modelling we obtain precise stellar properties of the triple, including the mass ratios ($M_ Aa /(M_ Ab1 +M_ Ab2 Ab1 /M_ Ab2 the separation ratio Aab /a_ Ab1Ab2 0.01$), orbital periods Aab Ab1Ab2 and stellar radii ( R R R Via radial velocity fitting and spectral disentangling, we find orbital elements for Aa, Ba, and Bb that are in excellent agreement with each other and with previous results in the literature. Spectral modelling on the disentangled spectrum of Aa provides constraints on the effective temperature eff K), surface gravity (log$(g) = 4.14 0.18$ dex), micro-turbulent velocity micro rotation velocity ($v i = 127 and metallicity M/H

Estimating the performance of wind turbines misaligned with respect to the wind direction using a simplified CFD based approach

Wind direction in atmospheric boundary layer changes constantly. As such, the power extracted from the wind turbines in a wind farm is subjected to change. This change in wind direction impacts the wake pattern which in turn affects the power output of the downstream wind turbines. Empirical wind farm models are useful in such cases to assess the overall performance of the wind farms. However, the results provided by these models can be made more useful if fluid-structure interaction (FSI) effects are also considered. In the present work, we begin by looking at the potential flow model and the issues associated with it and used a modified model with constant vorticity to analyze the wind power available to the downstream wind turbines in a wind farm misaligned with respect to wind direction for irrotational flow. A simplified FSI based approach is used to carry out the simulations without the need of modelling the complicated wind turbine geometry.

Semiclassical estimates for measure potentials on the real line

We prove an explicit weighted estimate for the semiclassical Schrödinger operator P = - h^{2} \partial^{2}_{x} + V(x;h) on L^{2}(\R) , with V(x;h) a finite signed measure, and where h >0 is the semiclassical parameter. The proof is a one-dimensional instance of the spherical energy method, which has been used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result is that the potential need not be absolutely continuous with respect to Lebesgue measure. Two consequences of the weighted estimate are the absence of positive eigenvalues for P , and a limiting absorption resolvent estimate with sharp h -dependence. The resolvent estimate implies exponential time-decay of the local energy for solutions to the corresponding wave equation with a compactly supported measure potential, provided there are no negative eigenvalues and no zero resonance, and provided the initial data have compact support.

NeuSIM4: A comprehensive GEANT4 based neutron simulation code

A new neutron SIMulation program based on the versatile GEANT4 toolkit, neuSIM4, has been developed to describe interactions of neutrons in the NE213 liquid scintillator from 0.1 to 3000 MeV. neuSIM4 is designed to accommodate complicated modern detector geometry setups with multiple scintillator detectors, each of which can be outfitted with more than one photo-multiplier. To address a broad spectrum of neutron energies, two new neutron interaction physics models, KSCIN and NxQMD, have been implemented in GEANT4. For neutrons with energy below 110 MeV, we incorporate a total of eleven neutron induced reaction channels on hydrogen and carbon nuclei, including nine carbon inelastic reaction channels, into KSCIN. Beyond 110 MeV, we implement a neutron induced reaction model, NxQMD, in GEANT4. We use its results as reference to evaluate other neutron-interaction physics models in GEANT4. We find that results from an existing cascade physics model (INCL) in GEANT4 agree very well with the results from NxQMD, and results from both codes agree with new and existing light response data. To connect KSCIN to NxQMD or INCL, we introduce a transition region where the contribution of neuSIM4 linearly decreases with corresponding increased contributions from NxQMD or INCL. To demonstrate the application of the new code, we simulate the light response and performance of a 2 × 2 m2 neutron detector wall array consisting of 25 2m-long scintillation bars. We are able to compare the predicted light response functions to the shape of the experimental response functions and calculate the efficiency of the neutron detector array for neutron energies up to 200 MeV. These simulation results will be pivotal for understanding the performance of modern neutron arrays with intricate geometries, especially in the measurements of neutron energy spectra in heavy-ion reactions.

Fourth-order exponential time differencing Runge–Kutta scheme and local meshless method to investigate unsteady diffusion–convection problems of anisotropic functionally graded materials

The current article is devoted to proposing a local meshfree method for simulating the unsteady diffusion–convection problems of anisotropic functionally graded materials (AFGM). The first stage of the proposed numerical algorithm is based on discretizing the spatial variables by utilizing the direct meshless local Petrov–Galerkin (DMLPG) method. Then a system of ODEs is extracted which is associated to the time variable. Furthermore, the fourth-order exponential time differencing Runge–Kutta method is applied to solve the system of ODEs. The main aim of this work is to introduce a flexible and low-cost numerical procedure to solve unsteady diffusion–convection problems of AFGM on the complicated geometry. The new numerical technique has admissible resultants to simulate the studied model. This numerical formulation is applied to several examples with complex domains to check its ability.

Sound pressure level determination around the transformer using coherent and incoherent analytical summation methods

In this paper, coherent and incoherent analytical summation methods from elementary sound sources on the transformer’s tank are presented. They aim to determine a one-third-octave spectrum of sound pressure levels (SPL) in points around the transformer. Each part of the transformer’s surface is treated as a separate plane source so the analytical calculation can be applied to find SPL in the surroundings. Reflections from the ground are also considered by putting an image of the plane sources on the other side of the reflecting plane and adding its contributions to the total pressure in the desired points. Grid-like vibration measurements of vibration velocity on the tank surfaces are used as input parameters. Vibrations and sound pressure levels are measured to validate the method on the 5 MVA transformer experimental object. The SPL around the transformer in short-circuit (SC) and open-circuit (OC) tests is measured in the semi-anechoic chamber to compare it with theoretical results. By analyzing the results, the coherent calculation with reflection provided the most accurate results. In the SC operating condition, the normalized root mean square error (NRMSE) is 17.2%, and in the OC operating condition, it is 10.9%. The novelty of the presented method is that it considers complicated transformer geometry where each surface is calculated as a separate noise source. It calculates noise at a distance from the elementary vibration sources, considering phase and reflection from the hard ground, and provides detailed noise maps that can be used for noise modeling around the substations.

Manipulation of Gaseous Ions with Acoustic Fields at Atmospheric Pressure.

The ability to controllably move gaseous ions is an essential aspect of ion-based spectrometry (e.g., mass spectrometry and ion mobility spectrometry) as well as materials processing. At higher pressures, ion motion is largely governed by diffusion and multiple collisions with neutral gas molecules. Thus, high-pressure ion optics based on electrostatics require large fields, radio frequency drives, complicated geometries, and/or partially transmissive grids that become contaminated. Here, we demonstrate that low-power standing acoustic waves can be used to guide, block, focus, and separate beams of ions akin to electrostatic ion optics. Ions preferentially travel through the static-pressure regions ("nodes") while neutral gas does not appear to be impacted by the acoustic field structure and continues along a straight trajectory. This acoustic ion manipulation (AIM) approach has broad implications for ion manipulation techniques at high pressure, while expanding our fundamental understanding of the behavior of ions in gases.

Dynamic wetting performance and resin-flow behavior of two-level selective laser-etched metal surface in resin transfer moulding process of fiber-metal laminates

ABSTRACT In selective laser etching for the interfacial strengthening of Fiber Metal Laminates (FMLs), the wettability of customized structured features significantly affects their overall mechanical properties. For two-level features, their complicated geometry makes the evaluation of wettability difficult, which introduces challenges to feature design. In this study, a sandwich FMLs comprising an AA2024-O metal skin and a plain-woven carbon-fiber fabric core is investigated. Three types of two-level feature interfaces are designed and introduced into the FMLs structure. To understand the effect of two-level features on the resin flow and dynamic wetting progress in resin transfer molding, two-phase flow Computational Fluid Dynamics numerical models coupled with porous media and level-set numerical methods are built. The feasibility of the numerical simulation in the qualitative analysis was verified by comparing the results with those of the testing porosity values from X-ray scanning. In addition, the mechanisms of pore formation and transportation are revealed. These results indicate that the deeper groove structure in the etched features plays an important role in discharging residual air and determining the transformation of pores to reduce porosity. The permeability orientations of the porous media and surface structure both have significant impacts on the flow behavior and formation of pores.

The evolution of continuum polarization in type II supernovae as a diagnostic of ejecta morphology

The linear polarization of the optical continuum of type II supernovae (SNe), together with its temporal evolution is a promising source of information about the large-scale geometry of their ejecta. To help access this information, we undertook 2D polarized radiative transfer calculations to map the possible landscape of type II SN continuum polarization (Pcont) from 20 to 300 days after explosion. Our simulations were based on crafted 2D axisymmetric ejecta constructed from 1D nonlocal thermodynamic equilibrium time-dependent radiative transfer calculations for the explosion of a red supergiant star. Following the approach used in our previous work on SN 2012aw, we considered a variety of bipolar explosions in which spherical symmetry is broken by material within ~30° of the poles that has a higher kinetic energy (up to a factor of two) and higher 56Ni abundance (up to a factor of about five, allowing for 56Ni at high velocity). Our set of eight 2D ejecta configurations produced considerable diversity in Pcont (λ ~ 7000 Å), although its maximum of 1–4% systematically occurs around the transition to the nebular phase. Before and after this transition, Pcont may be null, constant, rising, or decreasing, which is caused by the complex geometry of the depth-dependent density and ionization and also by optical depth effects. Our modest angle-dependent explosion energy can yield a Pcont of 0.5–1% at early times. Residual optical-depth effects can yield an angle-dependent SN brightness and constant polarization at nebular times. The observed values of Pcont tend to be lower than obtained here. This suggests that more complicated geometries with competing large-scale structures cancel the polarization. Extreme asymmetries seem to be excluded.

Harmonic oscillations of a piezoceramic functional-graded sectional cylinders with account of energy dissipation

Harmonic oscillations of piezoceramic functional-gradient sectioned hollow cylinders are studied, taking into account energy dissipation. A cylinder of finite length is considered, consisting of an even number of sections polarized in the circular direction, which are connected to each other by similar sides. The material is considered to be functionally heterogeneous in the direction of the previous polarization. The cylinder is loaded by the potential difference applied to the flat faces of the sections. The calculation is performed by the finite element method. The forming of damping matrix and determining the coefficients corresponding to the four ways of introducing Rayleigh damping into the FEM is described. Amplitude-frequency responses taking into account damping are built for displacements, charge on the electrodes, electrical admittance and its logarithm. The consideration of energy dissipation according to the Rayleigh damping model is related to the experimental data due to the Q factor of the piezo element. The frequencies of electrical resonances, anti-resonances and the corresponding electromechanical coupling coefficients for functionally inhomogeneous and homogeneous cylinders are determined. For the functional-gradient piezoelectric element of the considered configuration, the maximum CEMC occurs at the frequency of the second electrical resonance. At this frequency, the dynamic stress-strain state of the cylinder is investigated. Forms of oscillations and distribution of stress amplitude values are constructed. The largest normal stresses according to the Coulomb-Mohr strength theory are determined and compared with the von Mises strength theory. FEA allows three-dimensional calculation of harmonic oscillations of functional-gradient piezo elements of complicated geometry. The determined frequencies of electrical resonance and anti-resonance and the corresponding CEMC make it possible to choose the most effective operating mode of the piezo element. The FEM mathematical apparatus makes it possible to quickly and qualitatively evaluate the strength, determine the charge on the electrodes, the current in the circuit, build the amplitude-frequency characteristics of admittance and impedance, and evaluate the efficiency of energy conversion.