Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “variational analysis”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 793 records · Page 44
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

PINCH POINT ANALYSIS OF GAS COOLERS IN TRANSCRITICAL R1336mzz(Z) HIGH-TEMPERATURE HEAT PUMPS

The majority of high-temperature heat pumps (HTHPs) operate subcritical vapor compression cycles, where the maximum sink temperature is limited by the critical temperature of refrigerants. In HTHPs with a sink temperature of ≥ 150°C, transcritical cycles overcome this limitation by rejecting heat in the supercritical regions of refrigerant. Supercritical refrigerants in gas coolers have a large temperature glide and substantial nonlinear variation in specific heat capacity vs. temperature. Closely matching the refrigerant glide temperature and the heat sink temperature profile greatly impacts the heat transfer characteristic of gas coolers and the overall performance of HTHPs. This study conducts a pinch point analysis for the gas cooling process of supercritical R1336mzz(Z) in HTHPs. A finite volume element model of a supercritical R1336mzz(Z) counterflow gas cooler is built and solved. This study evaluates the influences of operating conditions, including inlet and outlet of coolants, refrigerant evaporation temperature and superheat degree, pinch temperature difference, and heat rejection pressure, on the performance of HTHPs. The optimal discharge pressures in gas coolers are proposed for industrial spray dryers in providing drying air at 170°C−210°C. The results of the present study help design gas coolers in transcritical R1336mzz(Z) HTHPs.

Wang, Pengtao↗
DOE OSTI2025

Preserving nonlinear constraints in variational flow filtering data assimilation

Data assimilation aims to estimate the states of a dynamical system by optimally combining sparse and noisy observations of the physical system with uncertain forecasts produced by a computational model. The states of many dynamical systems of interest obey nonlinear physical constraints, and the corresponding dynamics is confined to a certain sub-manifold of the state space. Standard data assimilation techniques applied to such systems yield posterior states lying outside the manifold, violating the physical constraints. This work focuses on particle flow filters which use stochastic differential equations to evolve state samples from a prior distribution to samples from an observation-informed posterior distribution. The variational Fokker-Planck (VFP)—a generic particle flow filtering framework—is extended to incorporate non-linear, equality state constraints in the analysis. To this end, two algorithmic approaches that modify the VFP stochastic differential equation are discussed: (i) VFPSTAB, to inexactly preserve constraints with the addition of a stabilizing drift term, and (ii) VFPDAE, to exactly preserve constraints by treating the VFP dynamics as a stochastic differential-algebraic equation (SDAE). Additionally, an implicit-explicit time integrator is developed to evolve the VFPDAE dynamics. The strength of the proposed approach for constraint preservation in data assimilation is demonstrated on three test problems: the double pendulum, Korteweg-de-Vries, and the incompressible Navier-Stokes equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Analysis of aerosol cloud interactions with a consistent signal of meteorology and other influencing parameters

Quantifying the impact of aerosols on cloud micro/macro physical properties and estimating the signature of Aerosol Cloud Interactions (ACI) is one of the challenging tasks in atmospheric sciences. The Moderate Resolution Imaging Spectroradiometer and the European Centre for Medium-Range Weather Forecasts ERA-5 reanalysis data are employed to systematically study the ACI over the monsoon region in Pakistan. Based on the monsoon occurrence and rainfall intensity, the whole region is divided into three sub-regions labeled as highly intensive (R1), moderately intensive (R2) and weak (R3) monsoon region. The results indicate that the monthly mean Aerosol Optical Depth (AOD) peaks in the summer monsoon months (Jun, Jul, Aug, Sep). Here, the well-known Twomey effect whereby the Cloud Droplet Radius (CDR) decreases with increasing AOD holds only over R3; the opposite effects (Anti-Twomey effect) are found over R1 and R2, all passing the test of statistical significance (p<0.05). The multi-year AOD is found to be positively correlated with Cloud Liquid Water Path (CLWP) and Cloud Optical Depth (COD) over R1 and R2, suggesting that thicker clouds containing more water droplets are formed in polluted atmosphere. Over R3, decreases in CLWP and COD are found with increasing AOD only when AOD is less than~0.325. The analysis of ACI over R1 and R2 during the winter months shows similar but stronger responses of CDR, CLWP and COD to the variation in AOD. The weaker responses during the summer monsoon season may attributed to the occurrence of high level cloud and unstable atmospheric condition. Further investigation of the influences of Relative Humidity and pressure vertical velocity on the CDR-AOD relationships shows that although the magnitude of the CDR-AOD correlations change with meteorological conditions, the sign of correlations remain unchanged with meteorological conditions.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Dissolution of periodic arrays of grains: Upscaling of pore-scale simulations with fast reactions

Upscaling methods are frequently used to derive transport equations at the macroscopic scale from more fundamental equations formulated at the pore scale. These methods typically give a suitable structure for the macroscopic equations and can also provide explicit expressions for the constitutive parameters, such as permeability and dispersion coefficients. Introducing chemical reactions complicates upscaling in at least two important ways. First, the interplay between chemical reactions and transport processes introduces a new length scale, which can be much smaller than the convective or dispersive length scales. A small reactive length scale breaks one of the key assumptions in upscaling; that there is a significant separation in length between the pore-scale and macro-scale processes. The second complication is that if reactions take place at mineral surfaces (dissolution or precipitation) then the pore space itself is evolving in time. In this paper we suggest ways in which these difficulties can be approached, based on analysis of pore-scale simulation data. First, we noticed that the concentration field in successive unit cells has an almost identical spatial variation, with a single scaling factor for each unit cell that is proportional to the incoming reactant flux. Using pore-scale simulations to determine the mass transfer coefficient in a few unit cells, we can calculate the concentration field in the whole domain, even when dissolution is entirely transport limited. Second, we have noticed a time-dependent mapping of the grain shapes from different unit cells. From these observations, we can deduce constitutive relations where the only time-varying parameter is the porosity. Here, we show that a model based on these ideas can quantitatively account for the pore-scale simulation data.

58 GEOSCIENCES↗
DOE OSTI2022

H&#x022E; 2 + H&#x022E; 2 : High level theory and the role of singlet channels

At low temperatures and high pressures, the H&#x022E; 2 + H&#x022E; 2 reaction is an important reaction in combustion. There are significant unresolved discrepancies between prior theoretical and experimental studies of this reaction. It has generally been presumed to occur as an abstraction on the triplet surface to produce H 2 O 2 + 3 O 2 . Here, we employ a combination of high-level electronic structure theory (the ANL0 composite method or multi-reference methods as appropriate), sophisticated transition state theory (vibrationally adiabatic torsions, variational, and variable reaction coordinate), and master equation analyses to predict the thermal kinetics on the H 2 O 4 surface. Notably, this analysis suggests a significant branching to H&#x022E; 3 + &#x022E;H near 1000 K via reaction on the singlet surface. This channel does not appear to have been considered in prior combustion models. H&#x022E; 3 itself is a metastable complex (bound by only 3 kcal mol -1 ) that rapidly dissociates to &#x022E;H + 3 O 2 . Thus, the net reaction for this channel, H&#x022E; 2 + H&#x022E; 2 → &#x022E;H + &#x022E;H + O 2 , converts two low reactivity H&#x022E; 2 radicals into two highly reactive &#x022E;H radicals. The ramifications of these newly derived rate expressions are highlighted through kinetic modeling studies of H 2 , CH 3 OH, $\textit{n}$-heptane, and isooctane oxidation; all at the high pressures of relevance to combustion devices.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Commercial aluminum powders, Part I: Particle size characterization and slow heating rate thermal analysis.

We characterized nine commercial aluminum (Al) powders using several methods to measure particle characteristics and thermal analysis, with the goal to understand how these parameters influence energy release. Although it is well-known that lot-to-lot variations in commercial nanoparticles are common, the Al powders were more heterogeneous than anticipated – both with regards to particle size distributions and impurities. Manufacturer specifications – often quoted in the literature without confirmation – were not always accurate for the specific sample lots we investigated. In several cases, different conclusions could be drawn from individual particle size techniques; a combination of multiple techniques provides a clearer picture of the powder properties. Thorough characterization of Al powders is required prior to interpretation of experimental results from a variety of applications. In particular, previous studies have shown contradictory results on the influence of Al on detonation performance, perhaps partially due to insufficient characterization of the Al powders themselves.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Mesoscale structural gradients in human tooth enamel

The outstanding mechanical and chemical properties of dental enamel emerge from its complex hierarchical architecture. An accurate, detailed multiscale model of the structure and composition of enamel is important for understanding lesion formation in tooth decay (dental caries), enamel development (amelogenesis) and associated pathologies (e.g., amelogenesis imperfecta or molar hypomineralization), and minimally invasive dentistry. Although features at length scales smaller than 100 nm (individual crystallites) and greater than 50 µm (multiple rods) are well understood, competing field of view and sampling considerations have hindered exploration of mesoscale features, i.e., at the level of single enamel rods and the interrod enamel (1 to 10 µm). Here, we combine synchrotron X-ray diffraction at submicrometer resolution, analysis of crystallite orientation distribution, and unsupervised machine learning to show that crystallographic parameters differ between rod head and rod tail/interrod enamel. This variation strongly suggests that crystallites in different microarchitectural domains also differ in their composition. Thus, we use a dilute linear model to predict the concentrations of minority ions in hydroxylapatite (Mg 2+ and CO 3 2- /Na + ) that plausibly explain the observed lattice parameter variations. While differences within samples are highly significant and of similar magnitude, absolute values and the sign of the effect for some crystallographic parameters show interindividual variation that warrants further investigation. In conclusion, by revealing additional complexity at the rod/interrod level of human enamel and leaving open the possibility of modulation across larger length scales, these results inform future investigations into mechanisms governing amelogenesis and introduce another feature to consider when modeling the mechanical and chemical performance of enamel.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Microbiome Comparison and Pathogen Identification for Three Migrating Passerines Captured During Spring Season in Jordan Using 16S rRNA Sequencing

Jordan is located on an important spot along the Mediterranean and Black Sea Flyway. Hundreds of migratory bird species have been identified stopping over in Jordan during spring and autumn migratory seasons. Compared to mammals and economically important birds, the microbiomes of wild bird species are severely understudied. Gut microbial composition is a valuable source of information that reflects food preferences, foraging behavior, and the risk of pathogen transmission to humans and other animals. In this study, we assessed the microbiome composition of three species of migrating passerines (willow warblers, lesser whitethroats, and common reed warblers) captured during the spring migration stopover in Jordan in 2023. A total of 59 fecal samples were selected evenly from the three species and subjected to 16S sequencing and microbiome analysis. Our objectives were to determine the diversity of bacteria in these three species, assess the amount of intra- and inter-specific variation, and detect pathogenic genera and species that could pose health risks to humans, domestic animals, and wildlife. Bacteria mainly belonged to the phyla Proteobacteria (62%), Actinobacteriota (18%), Firmicutes (13%), Cyanobacteria (5%), and Bacteroidota (1%). The results reveal that lesser whitethroats had the greatest variation in bacterial genus richness, Shannon diversity, and microbial composition compared to willow warblers and common reed warblers. The three bird species harbored several pathogenic genera and species, including Campylobacter, Enterococcus, Escherichia-Shigella, Mycoplasma, Rickettsia, Clostridium perfringens, and Vibrio cholerae. We suggest further investigation to understand the relationship between migratory behavior and their gut microbiome. We advocate for the use of advanced molecular techniques to characterize the pathogens found in migratory birds that might have public and environmental health impacts in addition to economic loss.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2021

Growth and characterization of large (Y,La)TiO 3 and (Y,Ca)TiO 3 single crystals

The Mott-insulating rare-earth titanates ( R TiO 3 , with R being a rare-earth ion) are an important class of materials that encompasses interesting spin-orbital phases as well as ferromagnet-antiferromagnet and insulator-metal transitions. The growth of these materials has been plagued by difficulties related to overoxidation, which arises from a strong tendency of Ti 3 + to oxidize to Ti 4 + . We describe our efforts to grow sizable single crystals of YTiO 3 , Y 1 - x La x TiO 3 ( x ≤ 0.25 ), and Y 1 - y Ca y TiO 3 ( y ≤ 0.35 ) with the optical traveling-solvent floating-zone technique. Finally, we present sample characterization via chemical composition analysis, magnetometry, charge transport, neutron scattering, x-ray absorption spectroscopy, and x-ray magnetic circular dichroism to understand macroscopic physical property variations associated with overoxidation. Furthermore, we demonstrate a good signal-to-noise ratio in inelastic magnetic neutron scattering measurements of spin-wave excitations. A superconducting impurity phase, found to appear in Ca-doped samples at high doping levels, is identified as TiO.

36 MATERIALS SCIENCE↗
DOE OSTI2020

PyMDA : microcrystal data assembly using Python

The recent developments at microdiffraction X-ray beamlines are making microcrystals of macromolecules appealing subjects for routine structural analysis. Microcrystal diffraction data collected at synchrotron microdiffraction beamlines may be radiation damaged with incomplete data per microcrystal and with unit-cell variations. A multi-stage data assembly method has previously been designed for microcrystal synchrotron crystallography. Here the strategy has been implemented as a Python program for microcrystal data assembly ( PyMDA ). PyMDA optimizes microcrystal data quality including weak anomalous signals through iterative crystal and frame rejections. Beyond microcrystals, PyMDA may be applicable for assembling data sets from larger crystals for improved data quality.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Design of a Single Layer Metamaterial for Pressure Offloading of Transtibial Amputees

While using a prosthesis, transtibial amputees may experience pain and discomfort brought on by large pressure gradients at the interface between the residual limb and prosthetic socket. Current prosthetic interface solutions attempt to alleviate these pressure gradients by using soft homogenous liners to reduce and distribute pressures. This research investigates an additively manufactured metamaterial inlay with a tailored mechanical response to reduce peak pressure gradients around the limb. The inlay uses a hyperelastic behaving metamaterial (US10244818) comprised of triangular pattern unit cells which can be 3D printed with walls of various thicknesses controlled by draft angles. The hyperelastic material properties are modeled using a Yeoh 3rd order model. The 3rd order coefficients can be adjusted and optimized, which corresponds to a change in the unit cell wall thickness to create an inlay that can meet the unique offloading needs of an amputee. Finite element analysis simulations evaluated the pressure gradient reduction from: 1) a common homogenous silicone liner, 2) a prosthetist's inlay prescription that utilizes three variations of the metamaterial, and 3) a metamaterial solution with optimized Yeoh 3rd order coefficients. When compared to a traditional homogenous silicone liner for two unique limb loading scenarios, the prosthetist prescribed inlay and the optimized material inlay can achieve equal or greater pressure gradient reduction capabilities. These results show the potential feasibility of implementing this metamaterial as a method of personalized medicine for transtibial amputees by creating a customizable interface solution to meet the unique performance needs of an individual patient.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2023

Iodine Air Sampling: Installation and Preliminary Results

The F-Area wetlands are a known sink for radioiodine generated at the Savannah River Site (SRS). Wastewater containing iodine-129 ( 129 I) was discharged into the F-Area Seepage Basins for disposal, but the acidity of the wastewater mobilized some of the contaminants from the basin soil through the vadose zone and into the groundwater; ultimately creating a groundwater plume that extends towards Fourmile Branch and its associated wetland areas. Dissolved radioiodine specifically has migrated to the nearby F-Area wetlands, where it persists as a contaminant of concern. Accumulation and resurfacing of 129 I in the wetlands and nearby areas creates a potential secondary source zone where changes in biogeochemical conditions could result in the release of this sequestered 129 I to the air. Monitoring data has indicated that 129 I has a seasonal behavior at several surface water stations, with high concentrations during summertime and low concentrations during wintertime. Still, it is not known which factors control the seasonal behavior of 129 I. Further, it has been reported that 129 I contributed approximately 23% to the 2019 SRS offsite air pathway dose to the representative person for that year (SRNS, 2020), but air measurements of 129 I have never been taken in the F-Area wetlands. Because air emissions of 129 I from the wetlands have never been investigated, it is not known if this could be a potential release pathway that needs to be included in the site conceptual model. To address these unknowns, the Savannah River National Laboratory (SRNL) has an applied research task to investigate whether 129 I in the wetlands is emitted to the air and, if so, whether air concentrations exhibit seasonal variation. This report presents information about the system recently installed in the F-Area wetlands for measuring 129 I, as well as analysis results for samples collected June through August 2023. Quantification of this constituent will provide a better understanding of the contribution of airborne 129 I from the F-Area wetlands to the offsite dose and better inform ongoing and future remedial actions.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Model for the Softening Factor within Stages of Work Hardening

A formulation for a dimensionless coefficient c b is derived that represents a scale of microstructural softening for alloys which follow Kocks-Mecking (K-M) work hardening behavior. The variation of the true plastic strain ε p between the proportional limit σ y and the strength σ u at the instability is determined using the Considère criterion. Parameterization of the model is limited to variables expressly measured within tensile experiments. Further development is now made for the softening factor c bi through the individual and sequential stages 3 and 4 of plastic deformation during K-M work hardening Q behavior. Application is shown for tensile test results of Ti-6Al-4V made by different additively manufactured (AM) processes. It is found that the variation in plastic strain as a function of c bi produces a continuous curve representative of the alloy system. The results of data analysis indicate that Θ o3 and c b3 increase, while Θ o4 and c b4 decrease, as the total plastic strain ε p increases. Furthermore, formulations derived for K-M stages 3 and 4 enable the evaluation of other material parameters such as the activation volume ν* for the onset of plastic deformation. This activation volume is found to be near constant, at a ν*-value of 0.353±0.036 nm 3 as computed using a strain-rate sensitivity of strength exponent m of 0.014, irrespective of the AM method used to produce the Ti-6Al-4V alloy.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Yield, chemistry, and TEA/LCA data for 84 switchgrass (Panicum virgatum L.) variants

Switchgrass yield, chemistry, technoeconomic assessment (TEA), and life cycle assessment (LCA) data referenced in the manuscript "Economics and Sustainability impacts of yield and composition variation in bioenergy crops: Panicum virgatum L.", Happs, et. al. (manuscript in revision at ACS Sustainable Chemistry & Engineering). These results are based on analysis of yield data generated from a set of switchgrass common garden experiments established by the Center for Bioenergy Innovation (CBI; https://cbi.ornl.gov/). These data include estimates of commercial-scale yields and characterization of biomass chemistry (fermentable carbohydrate fraction) for 84 different switchgrass variants from a genome-wide association study population, as well as estimates of minimum fuel selling price and various life-cycle footprint metrics for cellulosic ethanol produced from that switchgrass biomass.

switchgrass, biofuels↗
DOE OSTI2026

Data and Code for: Observation-constrained agroecosystem model inversion reveals continental-scale variation of winter wheat traits

This repository contains the simulation outputs and processing scripts associated with the study of winter wheat traits across the United States, utilizing the Ecosys agroecosystem model. The dataset includes model results for both rainfed and irrigated winter wheat systems, supporting the findings presented in the manuscript titled "Observation-constrained agroecosystem model inversion reveals continental-scale variation of winter wheat traits." Data includes the original Ecosys simulation outputs (archived in .db format within the compressed .zip files) and extracted analysis data (stored in .pkl files for efficient processing). Python code for data processing and figure generation is provided in a Jupyter notebook. External Observational Datasets should refer to the following official repositories for the input and validation data used in this study. The eddy covariance data from the AmeriFlux network (https://ameriflux.lbl.gov/). Climate-forcing data of NLDAS-2 from NASA LDAS (https://ldas.gsfc.nasa.gov/nldas/nldas-2-forcing-data). Soil data from the Gridded Soil Survey Geographic Database (gSSURGO), available at (https://www.nrcs.usda.gov/resources/data-and-reports/gridded-soil-survey-geographic-gssurgo-database). Crop yields, planting and harvest dates from the USDA public databases (https://quickstats.nass.usda.gov/; https://webapp.rma.usda.gov/apps/actuarialinformationbrowser/CropCriteria.aspx). Satellite-derived SLOPE GPP data from ORNL DAAC (https://daac.ornl.gov/cgi-bin/dsviewer.pl?ds_id=1786). Land use and crop progress information from the USDA Crop Data Layer and Crop Progress and Condition Gridded Layers (https://www.nass.usda.gov/Research_and_Science/). The Ecosys model code is available online at https://github.com/jinyun1tang/ECOSYS.

Wheat↗
DOE OSTI2026

Dataset of Generative AI Workload Power Profiles

This dataset provides a collection of high-resolution (5/10 Hz or every 0.2/0.1 seconds) power consumption profiles for generative artificial intelligence (GenAI) workloads executed on NLR's High Performance Computing (HPC) platform Kestrel. The dataset also includes examples of representative whole-facility power profiles generated using a bottom-up, event-driven, data center energy model . This dataset is designed to support research in energy modeling, infrastructure planning, energy system integration, and sustainability analysis for AI-driven computing systems. The dataset captures time-resolved electrical power measurements across a diverse set of configurations, including variations in job type (inference vs. training), workload (LLM vs. image generation), datasets, and number of compute nodes. Power traces are provided in a standardized format and include both raw/instantaneous and aggregated files. Each profile is accompanied by metadata describing workload parameters, enabling reproducibility and cross-study comparison. The dataset is intended for use in applications such as data center infrastructure planning, energy modeling, demand response and grid impact studies, and development and validation of system-level simulation tools. By making these workload-specific power profiles publicly available, this dataset aims to address the current lack of open, empirical energy data for generative AI systems and to facilitate transparent, reproducible research on the energy and environmental impacts of large-scale AI deployment. If you use this dataset, please cite the associated publication: Vercellino et al., “Measurement of Generative AI Workload Power Profiles for Whole-Facility Data Center Infrastructure Planning,” arXiv:2604.07345 (2026).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Elastic Bayesian Model Calibration

Functional data are ubiquitous in scientific modeling. For instance, quantities of interest are modeled as functions of time, space, energy, density, etc. Uncertainty quantification methods for computer models with functional response have resulted in tools for emulation, sensitivity analysis, and calibration that are widely used. However, many of these tools do not perform well when the computer model’s parameters control both the amplitude variation of the functional output and its alignment (or phase variation). This paper introduces a framework for Bayesian model calibration when the model responses are misaligned functional data. The approach generates two types of data out of the misaligned functional responses: (1) aligned functions so that the amplitude variation is isolated and (2) warping functions that isolate the phase variation. These two types of data are created for the computer simulation data (both of which may be emulated) and the experimental data. The calibration approach uses both types so that it seeks to match both the amplitude and phase of the experimental data. The framework is careful to respect constraints that arise, especially when modeling phase variation, and is framed in a way that it can be done with readily available calibration software. In conclusion, we demonstrate the techniques on two simulated data examples and on two dynamic material science problems: a strength model calibration using flyer plate experiments and an equation of state model calibration using experiments performed on the Sandia National Laboratories’ Z-machine.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Sensitivity analysis of a layered piezoelectric system using ZFEM

The complex variable finite element method (ZFEM) is a numerical technique which aims to find the partial derivatives of the independent variables with respect to variation in dependent parameters declared in the physics. This is done by combining the complex Taylor series expansion within the weak formulation of the governing equation in a coupled system of linear equations forming a complex valued block matrix given by the Cauchy–Riemann matrix representation. In this work, two-dimensional linear first-order elements have been implemented in ZFEM to predict the design derivatives of the mechanical displacement field and the voltage potential field for a layered piezoelectric system in a steady-state study with Dirichlet boundary condition applied at the top and bottom edges of the geometry. This approach allows the standard FEM solution to quantify the sensitivity of the mechanical displacement and voltage potential fields with respect to small variations in the material properties through the information obtained from the computation of the derivatives. The domain is formed by a layered body with PZT-4 and PZT-5 stacked together. For result verification, the numerical solution obtained with ZFEM was compared to results from a commercial FEM package and the solution from the imaginary part was compared to the exact solution of a well-known benchmark problem. In conclusion, comparison of the results showed good agreement for both the real and imaginary parts of the solution and the largest sensitivities were found in PZT-5 specifically in C 13 , C 33 , and ε 33 .

42 ENGINEERING↗