Radiation Boundary Conditions Research Articles

Study on the Characteristics of Trackside Acoustic Flow Field of High-Speed Train under the Influence of Crosswind.

During the on-track acoustic detection process, a potential flow model and an acoustic finite element mathematical model based on synthetic wind are utilized, taking into account the combined effects of vehicle speed, wind direction angle, and crosswind speed. Simulation and modeling are achieved using Automatic Matching of Acoustic Radiation Boundary Conditions (AML) technology, enabling obtaining a distribution map and sound pressure frequency response curve of the trackside acoustic field under crosswind conditions by setting up field point grids. It is found that sound pressure values at the same location gradually increase as the vehicle speed increases in the frequency range of 10 Hz to 70 Hz, at different vehicle speeds. The sound pressure values and distribution area of the trackside acoustic field are the largest when the crosswind speed is 10 m/s (wind force at level five), allowing for easier location of the sound source when a fault occurs. The study also reveals that under different wind direction angles, the same location's sound pressure value on the trackside gradually decreases as the wind direction angle increases, to lower than that of the non-crosswind condition, severely hindering the reception and diagnosis of acoustic signals.

A novel technique for implementing the finite element method in a shallow water equation

We presented a novel approach to investigate the two-dimensional shallow water equation in its primitive form. Its employs the P1NC−P1 element pair to simulate various cases: standing waves, dam-break planar, and wave absorbing with embedded radiation boundary conditions. Unlike the conventional method, we approximate the free surface variable using a conformal basis P1 whereas the velocity potential is approximated using a non-conformal basis, P1NC. Thus, for each case, the weak form needs to be reformulated as well as the discrete form. The resulting scheme is a first-order ordinary differential system and solved by Crank Nicholson. The mass matrix in the momentum equation contains the multiplication between the two bases, which computed by the mass lumping. So, our method is explicit, flexible and easy to implement. Validation using standing waves demonstrated first-order accuracy, free from numerical damping and convergent to the analytical solution. Dam-break simulation result shown an agreement with ANUGA software. Our scheme's flexibility is demonstrated when it can mimic wave absorbing simulation employing embedded radiation boundary conditions. The reflection at the boundary seems small enough, thus can be neglected. All these findings have shown the robustness and capability of our scheme to predict accurate results for various shallow water flow problems.•A novel technique for solving 2D SWE in primitive form•It is explicit, flexible, easy to implement, accurate, and robust•Our approach is suitable for coastal/oceanographic simulations

Computational analysis of radiative nanofluid flow past an inclined cylinder with slip effects using the Yamada–Ota model

The notion of the paper is to analyse the flow attributes of a carbon nanotube-based hybrid nanofluid over an inclined cylinder. This study accounts for the impacts of mixed convection, thermal radiation, suction/blowing, and slip boundary conditions. The solution of the governing equations is obtained using the "bvp4c solver", and presented through graphs and tables. The obtained results are validated by comparing them with published work. The study reports that increasing the temperature slip parameter led to a reduction in both velocity and temperature distributions. The presence of an electric parameter contributed to an increase in nanofluid flow, while the temperature of the nanofluid exhibited a decrease. Moreover, increasing the Hartmann number from 0.2 to 1.1 results in a 1.39% increase in the heat transfer rate within the hybrid nanofluid model. However, the pure fluid model demonstrates a more substantial enhancement, with a significant 3.8% rise in the heat transfer rate.

Computational investigation of magnetized hybrid nanofluids heat transport and flow through elongational surface with thermal radiation and wall slip

The improved thermal performance of recently discovered hybridized nanofluids has become essential in large scale thermal processes. In fact, this is highly efficient technique to introduce the thermal efficiency of tranditional heat transferring fluids. The behavior of the nanofluid can be significantly impacted by the unsteady heating and magnetic field effects that may be present in many applications. Therefore, the current study investigat the unsteady magnetized flow of hybrid nanofluid with heat transport characteristics subject to thermal radiation and slip at the surface wall. The shrinking/stretching surface is chosen as a flow source, which is frequently occure in polymer technology, which deals with the deformability of elastic sheets, and in metallurgy, where continued strips are cooled. The novel form of shrinking surface flow is fundamentally a reverse flow and exhibits physical characteristics that differ significantly from the channel flow scenario. The distinctive features of this scruinity is the use of empirical relations to approximate the optimum thermophysical attributes of a Cu−Al2O3/ water hybrid nanofluid in order to model the 2-dimensional flow past a flat shrinking/stretching sheet under the action of radiation, Lorentz forces and realastic boundary condition responses. The governing system of modelled equation are assembled using the Tiwari-Das model in conjunction with a hybrid mass-based nanofluid model. The bvp4c algorithm is employed within the computer MATLAB programme. The hybrid nanofluid flow shows conclusive improvement in the frictional coefficient and heat transport performance. However, the effectiveness the unsteadiness parameter deteriorates the heat transmission. In the contiguity of a suction parameter, multiple outcomes appear to arise for both stretched and shrinking instances. The coefficient of energy transport improves as the magnetic factor is augmented, however the skin coefficient of friction exhibits dual behavior for the second solutions. A time-dependence investigation is undertaken to figure out the reliability of the twin solutions, and it is discovered that merely one of them remains stable and aesthetically credible.

A solution for the neutron diffusion equation in the spherical and hemispherical reactors using the residual power series

A novel analytical solution to the neutron diffusion equation is proposed in this study using the residual power series approach for both spherical and hemispherical fissile material reactors. Various boundary conditions are investigated, including zero flux on the boundary, zero flux on the extrapolated boundary distance, and the radiation boundary condition (RBC). The study also shows how two hemispheres with opposing flat faces interact. We give numerical results for the same energy neutrons diffused in pure P239u. By qualitative comparison with the homotopy perturbation method and Bessel function-based solutions, the residual power series method (RPSM) presents accurate series solutions that converge to the exact solutions, as shown in this study. Moreover, numerical results were shown to be improved by the computer implementation of the analytic solutions.

The Role of Quadratic-Linearly Radiating Heat Source with Carreau Nanofluid and Exponential Space-Dependent Past a Cone and a Wedge: A Medical Engineering Application and Renewable Energy

A mathematical analysis for a steady, incompressible, laminar Carreau nanofluid flow over a porous cone and wedge is considered. The novelty of this research work is to scrutinize an exponential space-dependent Cattaneo–Christov heat flux and quadratic Rosseland approximation effects. Two-phase nanofluid model, i.e., Brownian motion, and thermophoretic effects are included. This study also employs non-linear thermal radiation and convective surface boundary conditions to investigate heat transport phenomena. The Carreau nanofluid boundary layer equations are derived using the standard boundary layer approximations. By applying the similarity transformations, the managing set of partial differential equations (PDEs) becomes a system of connected non-linear ordinary differential equations (ODEs). The resultant non-linear ODEs are solved numerically utilizing the numerical method, particularly the bvp4c function in MATLAB. The significances of the fluid motion are visualized by sketching several Carreau nanofluid flow’s relevant parameters using graphic outputs. The effects of a wide range of thermophysical variables on liquid properties including velocity, temperature, concentration, skin-friction, Nusselt number, and Sherwood number are investigated and discussed. This study’s findings are compared to previously announced results within a limited range, where strong validation was seen. The results demonstrate that the opposite mechanism is observed with repercussion of Weissenberg number (We) on velocity profile and concentration, the enhancement of We increases the momentum boundary layer while it reduces the concentration boundary layer. The outcomes could be applied to the cooling of equipment, electronics, and various industrial units.

Enhanced heat transfer analysis of hybrid nanofluid over a Riga plate: Incorporating Lorentz forces and entropy generation

The Riga plate, comprising a configuration of electrodes and permanent magnets, can induce a wall-parallel Lorentz force through external electric and magnetic fields, enabling effective control of fluid flow. This paper focuses on studying the flow characteristics of Al2O3-Cu/water hybrid nanofluid past a Riga plate in the presence of strong suction. The mathematical model incorporates a two-phase approach to describe the behavior of the nanofluid, while considering the Grinberg-term to account for the wall parallel Lorentz force generated by the Riga plate. Several influential factors such as entropy generation, viscous dissipation, heat flux, thermal radiation, slippage, and convective boundary conditions of temperature are considered in our investigation. Entropy generation within thermodynamic systems contributes to energy losses resulting from various factors, including frictional forces, viscosity, and chemical reactions. Minimizing entropy generation is crucial to enhancing system performance. To address the research problem, we utilize dimensionless parameters and assume strong suction to simplify the governing partial differential equations. The dimensionless model is subsequently solved numerically using the Adams-Bashforth technique. Additionally, a comprehensive graphical analysis is conducted to examine the impact of various dimensionless parameters on the flow fields. This study highlights the importance of controlling fluid flow using the Riga plate in the presence of suction and investigates the behavior of a specific nanofluid under these conditions. The results provide valuable insights into the influence of key parameters on the flow characteristics. We conclude that effective utilization of the Riga plate can lead to significant enhancements in fluid flow control. Moreover, the study contributes to reducing energy losses through minimizing entropy generation. The novelty of this work lies in its comprehensive approach, considering multiple influential factors and employing a two-phase model for the nanofluid. Additionally, the inclusion of the Grinberg-term to account for the wall parallel Lorentz force adds to the uniqueness of the study. This research expands upon previous efforts in the literature by providing a more comprehensive analysis of the fluid flow control using the Riga plate, incorporating a broader range of factors, and presenting a numerical solution approach.

A finite element approach for nonlinear, transient heat conduction problems with convection, radiation or contact boundary conditions

Heat conduction problems with convection and/or radiation boundary conditions arise in nuclear and other engineering fields. This paper presents a finite element approach to solve transient, three-dimensional (3-D), nonlinear heat conduction problems using quadratic tetrahedral elements with area/volume coordinates and quadrature integrations. Boundary conditions modeled include surface convection, to-ambient radiation, and surface-to-surface radiation (using the radiosity matrix method, RMM). Instead of the commonly used hemi-cube method, L′Huilier’s theorem is used here to calculate view factors. Contact constraints are also imposed. Interface condition is set up for the interfaces of different material. The one- and two-step backward differentiation are used for time integration. Nonlinearities introduced by temperature-dependent material thermal properties and the Stefan–Boltzmann law for radiation are handled using the Picard and/or Newton–Raphson methods. The convergence rate of the numerical scheme is verified using an analytical solution obtained using the method of manufactured solutions (MMS). Several practical cases are simulated, including that of two spheres in contact with each other, and results are compared to those obtained using commercial codes.

Momentum and thermal transport analysis in MHD nanofluid through the thermally heated square conduit: Finite element method

ObjectiveThe purpose of this investigation is to report the thermal performance of water-based nanofluids with two different types of hybrid nanofluids for the cooling process of the radiator inside the thermally heated square cavity under the addition of thermal radiation and magnetic field by using the finite element method. Research gapThe heat transfer analysis of MHD nanofluid with the suspension of titanium alloy Ti6A14V and aluminum alloy AA7075 in cavity flow problems under the effects of thermal radiation and thermal boundary conditions by using the Galerkin finite element method is not reported before. NoveltyIn this numerical investigation, the novel mathematical model of two types of titanium alloy Ti6A14V (class of titanium) and aluminum alloy AA7075 is proposed first time in the cavity flows under the influence of a uniform magnetic field and non-uniform thermally heated bottom wall. The momentum and energy equations are updated with terms of magnetic field and thermal radiation, respectively. The cavity is filled with AA7075- Ti6A14V/water nanofluids. Two parallel vertical walls of the cavity are set at constant temperatures (Tc) and the top one is adiabatic while the temperature at the lower wall is taken as the variable. Method of solutionThe physical model is composed in terms of nonlinear differential equations and transformed into a dimensionless form by using suitable non-dimensional quantities. The Galerkin finite element method along with iterative Newton’s Raphson method is selected to simulate the problem and obtain the numerical solution with a high degree of accuracy. The results are illustrated for fixed Prandtl number Pr=10.0 throughout the calculation, Rayleigh number 1000≤Ra≤100000, nanoparticle volume friction 0.0≤ϕ≤0.05 Hartman number 0≤Ha≤10 and thermal radiation parameter 1.0≤Ra≤10.0. Both grid independence tests and comparison tests are directed to get the accuracy of the established model. Important findingsThe computational results revealed that the aluminum alloy (AA7075) alloys nanoparticles are more productive as compared to titanium (Ti6Al4V) alloy nanoparticles in terms of enhancing the thermal features of water-based nanofluid. The aluminum alloy nanofluid and titanium alloy nanofluid provide 3.9 % & 5.7 % and 2.9 % & 4.2 % more heat transfer at the bottom and side walls as compared to regular fluid. ApplicationsThe current nanofluid models are more efficient than the conventional model to control the heat transfer rate and flow phenomena. The computational results revealed that if the radiator is assembled by using the modern concept of hybrid nanofluid then that will increase the heat transfer of the engine and enhance the lifespan of the machinery.

A study of the radiation thermal boundary conditions influence in three-dimensional thermomechanical contact problems

Any body, by the simple fact of having a certain temperature on its surface, emits heat in the form of electromagnetic radiation. The heat flux emitted is proportional to the fourth power of the temperature of the solid, increasing the inherent non-linearity of the thermomechanical problems. This energy transfer, for systems at high temperatures, has greater relevance than the possible exchange of heat between bodies in contact by conduction or convection processes with a surrounding fluid in the interstitial areas. For this reason, this work presents a robust formulation to study three-dimensional thermomechanical contact problems where radiation boundary and interstitial conditions are simulated. The Boundary Element Method is used to compute the thermomechanical influence coefficients, and a double iterative procedure is used to ensure the achievement of the thermomechanical contact conditions. The final non-linear thermomechanical system of equations is solved using a Newton-Raphson procedure. The proposed formulation is validated with a comparison with some benchmark problems found in the literature and afterward applied to solve different three-dimensional thermomechanical contact problems where the thermomechanical variables are greatly affected by radiation boundary and interstitial conditions.

Explicitly determined sea ice emissivity and emission temperature over the Arctic for surface‐sensitive microwave channels

AbstractData assimilation of satellite microwave measurements is one of the important keys to improving weather forecasting over the Arctic region. However, the use of surface‐sensitive microwave‐sounding channel measurements for data assimilation or retrieval has been limited, especially during winter, due to the poorly constrained sea ice emissivity. In this study, aiming at more use of those channel measurements in the data assimilation, we propose an explicit method for specifying the surface radiative boundary conditions (namely emissivity and emitting layer temperature of snow and ice). These were explicitly determined with a radiative transfer model for snow and ice and with snow/ice physical parameters (i.e. snow/ice depths and vertical distributions of temperature, density, salinity, and grain size) simulated from the thermodynamically driven snow/ice growth model. We conducted 1D‐Var experiments in order to examine whether this approach can help to use the surface‐sensitive microwave temperature channel measurements over the Arctic sea ice region for data assimilation. Results show that (1) the surface‐sensitive microwave channels can be used in the 1D‐Var retrieval, and (2) the specification of the radiative boundary condition at the surface using the snow/sea ice emission model can significantly improve the atmospheric temperature retrieval, especially in the lower troposphere (500 hPa to surface). The successful retrieval suggests that useful information can be extracted from surface‐sensitive microwave‐sounding channel radiances over sea ice surfaces through the explicit determination of snow/ice emissivity and emitting layer temperature.

Impact of Moving/Exponentially Accelerated Vertical Plate on Unsteady Flow and Heat Transfer in Hybrid Nanofluids

The main goal of this work is to explore exact analytical solutions for the transient hybrid nanofluid flow with heat transfer owing to a moving/exponentially accelerating infinite flat vertical plate with heat flux boundary conditions. Further, the uniqueness of this work is to investigate the impact of different types of hybrid nanofluids on heat transfer and unsteady flow features in the existence of thermal radiation and heat flux boundary conditions. For engineering variables like Nusselt number and skin friction coefficient, along with temperature and velocity profiles, graphs are used to reveal the results of the Laplace transform method. Increased heat transfer and friction values have been found for an exponentially accelerating plate. The findings can be utilized in heat exchangers as well as in electronics and chemical and biological reactors.

Semi-Analytical Solution for Modelling Moving Heat Sources in a Semi-Infinite Medium with Radiative and Convective Boundary Conditions

The weld quality is highly related to the thermal history of the weld and there have been many trials to monitor the quality using an infrared (IR) sensor. To obtain the real temperature of a surface based on the brightness temperature values measured by an IR camera, the emissivity value must be derived. For an accurate assessment of the emissivity, one must be aware of the melting point isotherm. The temperature profiles only depend on three factors during laser processing, specified as constants the characteristics of the material: laser beam speed (v), laser beam diameter (d), and power (P). Predicting the width of the melted zone reached during the welding process as the parameters vary is a tool for helping a quality laser processing and for determination of true temperature in laser welding using IR camera. This study describes the semi-analytical (SA) solution of the heat conduction equation for a localized moving Gaussian heat source with constant parameters on a semi-infinite medium. The solution, simple and quick to obtain, provides information on the width of the melted zone with an average error < 5 %. The outcome is assessed numerically and contrasted with FEM solutions for a Gaussian source, the latter having undergone experimental validation. With two distinct defocus values, def0 and def-6, and by varying the speed and power settings, two separate types of experiments were run. Thus, the SA solution was obtained and compared after the FEM solution had been obtained with a good approximation (max err 4.3 %, average err 2.7 %). Only in regard to the 1AL test is an error more than 5 % detected; in the other case, the average error is 3.75 %. Two more tests at the defocus values of def-4 and def-8 were conducted to confirm the model's validity as the parameters varied. Overall, the average error between the semi-analytical and the FEM solution is 4.1%. The SA solution may be used to effectively estimate the isotherms related to the melting point of aluminum (770 K). This allows to obtain a tool which helps restoring the real temperature based on the brightness values measured by the IR camera during laser welding. At the same time, this effective tool allows to investigate the importance of different processing parameters in laser manufacturing.

Automatic CHIEF Point Selection for Finite Element–Boundary Element Acoustic Backscattering

Computing the backscattering of harmonic acoustic waves from underwater elastic targets of arbitrary shapes is a challenging problem of considerable practical significance. The finite element method is well suited for the discretization of the target, while the boundary element method addresses the radiation boundary condition at infinity. A disadvantage of the boundary integral method is that it yields non-unique solutions at certain wavenumbers. This failure is associated with the existence of eigensolutions of the Helmholtz equation in the interior of the complement of the fluid domain (acoustic modes). The combined Helmholtz integral equation formulation (CHIEF) credited to Schenk is employed to combine the surface Helmholtz boundary integral with equations of the interior Helmholtz relation written down at selected points within the cavity of the scatterer (i.e., in the complement of the fluid domain).The difficulty associated with this approach has always been the lack of guidance on the necessary number of interior points and on their locations. The solution to this problem proposed here is to compute the acoustic modes using the finite element method to complement of the fluid domain and to identify locations of the peaks.This novel approach aids the decision as to how many points should be employed and where they should be located. Our numerical experiments demonstrate the robustness of the proposed automatic selection of the CHIEF points’ numbers and locations.

Daily INSOLation (DINSOL-v1.0): an intuitive tool for classrooms and specifying solar radiation boundary conditions

Abstract. Climate modelling requires spending an extensive amount of time programming, which means reading, learning, testing, and evaluating source code. Fortunately, many climate models have been developed within the past decades, making it easier for climate studies to be conducted on a global scale. However, some climate models have millions of code lines, making the introduction of new parameterizations a laborious task that demands teamwork. While it is true that the high-complexity models perform realistic climate simulations, some researchers perform their studies using simplified climate models in the preliminary test phases. This realization motivated the development of Daily INSOLation (DINSOL-v1.0), a robust computer program to support the simplified climate models, performing solar radiation calculations while considering Milankovitch cycles and offering various simulation options for its users. DINSOL was intended to function as a program that supplies data (e.g. daily insolation, instantaneous solar radiation, orbital parameters of the Earth, and calendar dates), such as the Paleoclimate Modelling Intercomparison Project (PMIP). While preparing the boundary conditions of solar radiation for climate models, it was realized that the DINSOL model could also be a helpful tool for use in classrooms. Thus, it was decided that an intuitive graphical user interface would be required to cater to this educational purpose. The model was written in the Fortran 90 language, while its graphical user interface would be built using PyGTK, a Python application programming interface (API) based on GIMP ToolKit (GTK). Furthermore, the R language would also be used to generate a panel containing contour fields and sketches of the orbital parameters to support the graphical execution. The model evaluation made use of data from PMIP and other tools, and the data analysis was performed through statistical methods. Once all tests were concluded, an insignificant difference between the DINSOL-obtained results and the results obtained from other models validated the viability of DINSOL as a dependable tool.

Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman–Kac Approach

We derive, through subordination techniques, a generalized Feynman–Kac equation in the form of a time fractional Schrödinger equation. We relate such equation to a functional which we name the subordinated local time. We demonstrate through a stochastic treatment how this generalized Feynman–Kac equation describes subdiffusive processes with reactions. In this interpretation, the subordinated local time represents the number of times a specific spatial point is reached, with the amount of time spent there being immaterial. This distinction provides a practical advance due to the potential long waiting time nature of subdiffusive processes. The subordinated local time is used to formulate a probabilistic understanding of subdiffusion with reactions, leading to the well known radiation boundary condition. We demonstrate the equivalence between the generalized Feynman–Kac equation with a reflecting boundary and the fractional diffusion equation with a radiation boundary. We solve the former and find the first-reaction probability density in analytic form in the time domain, in terms of the Wright function. We are also able to find the survival probability and subordinated local time density analytically. These results are validated by stochastic simulations that use the subordinated local time description of subdiffusion in the presence of reactions.

Numerical Investigation of Film Coefficient Approximation for Chemically Reacting Boundary-Layer Flows

Aerothermal analysis of spacecraft planetary entry is heavily dependent on heritage engineering models. The film coefficient heat transfer model examined in this paper estimates the convective heating to the vehicle for a laminar, dissociated, chemically reacting boundary layer for an Earth atmosphere. This model requires information about the vehicle and flowfield for a given trajectory point and estimates a proportional relationship between enthalpy potential and convective heat flux. In practice it is the aerothermal engineer who must decide which assumptions are appropriate for his/her application. This work looks at numerous CFD simulations for an arbitrary, axisymmetric flight vehicle to analyze the relative importance of both the mass and energy constraints imposed at the wall boundary, as well as the effect of various diffusion models. Within the subset of tested energy boundary conditions, it is found that the most desirable energy boundary condition is the radiative equilibrium boundary condition, which permits conservative estimates of convective heat flux, but also generates flowfield-dependent spatial thermal distributions along the surface. Other key findings are presented in an effort to make the film coefficient engineering model readily available to design engineers across industry.

Application of macros for automated performance of the same type operations when simulating SIW-based slotted waveguide antennas in Ansys HFSS CAD

SIW slotted waveguide antennas contain numerous equidistantly spaced vias and many slots distributed following a particular field distribution. Ansys HFSS CAD user must manually repeat the same type of operations when creating slots or vias. Automating these routine operations is advisable to eliminate design errors and speed up model construction. This paper aims at complete automation of SIW resonant slotted waveguide antenna arrays modelling in CAD Ansys HFSS, including model construction, running simulations and displaying simulation results. There are multiple procedures for slotted waveguide antenna synthesis. For small waveguide antenna arrays with a number of elements less than 15, the internal and external slots coupling consideration is beneficial, while for large waveguide antennas, no mutual coupling assumption can be made (so-called “energy method”). The energy method has been used to determine the coordinates of the longitudinal slots on the broad wall of the SIW based on a given side lobe level. The slot resonant length has been determined using substrate permittivity and operating frequency. The number of slots has been obtained based on the required half-power beamwidth. SIW dimensions have been calculated using the given operating frequency and the dimensions of the filled rectangular waveguide. An IronPython script has been developed to automate the antenna's model construction, automatic simulation and determination of its characteristics. All script commands have been taken from the “HFSS Scripting Guide” document integrated into Ansys HFSS. The tangible outcome of this work is the development of the IronPython script for fully automated modelling of SIW resonant slotted waveguide antennas. The script builds an antenna model from scratch based on a given antenna characteristics, assigns excitation port and radiation boundary conditions, and displays the simulation results – a radiation pattern at a fixed frequency. The script significantly reduces the antenna design time by eliminating the need to repeat the object creation operations.

Time domain response of a piezoelectric layer

A number of one-dimensional models have been developed to inform the design of piezoelectric transducers. The majority of these models are in the frequency domain. In this paper, we develop a one-dimensional time-domain model for the mechanical response of a piezoelectric layer. Secondary effects, resulting from feedback between the acoustic and electric variables, are included in the model. Our approach utilizes Green's function for the Helmholtz equation with radiation boundary conditions and the methods of complex analysis. The model predictions are validated by comparison with a finite-difference time-domain numerical simulation of the governing acoustic equations in and outside the layer. This time-domain model enables efficient calculation of the secondary piezoelectric action effects and provides the mechanical response to an arbitrary electrical source.

Complete radiation boundary conditions for the Helmholtz equation II: domains with corners

Complete radiation boundary conditions for the Helmholtz equation II: domains with corners