Prescribed Boundary Conditions Research Articles

A non-boundary-fitted-grid method, based on compact integrated-RBF approximations, for solving differential problems in multiply-connected domains

This paper presents a new non-boundary-fitted-grid numerical technique for solving partial differential equations (PDEs) in multi-hole domains. A multiply-connected domain is converted into a simply-connected domain of rectangular or non-rectangular shape that is discretised using a Cartesian grid. Compact radial basis function (RBF) stencils, which are constructed through integration rather than the conventional differentiation, are used to discretise the field variables. The imposition of inner boundary conditions is conducted by means of body forces that are derived from satisfying the governing equations and prescribed boundary conditions in small subregions. Salient features of the proposed method include: (i) simple pre-processing (Cartesian grid), (ii) high rates of convergence of the solution accuracy with respect to grid refinement achieved with compact integrated-RBF stencils, where both nodal function and derivative values are included in the approximations, (iii) the system matrix kept unchanged for the case of moving holes, and (iv) no interpolation between Lagrange and Euler meshes required. Several linear and nonlinear problems, including rotating-cylinder flows and buoyancy-driven flows in eccentric and concentric annuli, are simulated to verify the proposed technique.

Equivalent micropolar model for porous guided bone regeneration mesh: Optimum design for desired mechanical properties

In the present work, a micropolar continuum model is adopted to homogenise a heterogeneous porous model of Guided Bone Regeneration (GBR) meshes. GBR meshes are used in dentistry as mechanical barriers to isolate and protect the area of bone loss from the surrounding tissue while allowing new bone growth. The mechanical constants of the continuum are derived based on the strain energy equivalence of a periodic porous plate with its equivalent ortho-tetragonal micropolar model under prescribed boundary conditions. The effects of various architectural features such as pore shapes, patterns and sizes on the material parameters are investigated. The results show that the micropolar theory provides a better prediction of the response of the 2D porous geometries considered for the GBR mesh, compared to the classical Cauchy theory. The collected equivalent material parameters are further used for GBR mesh design, considering both mechanical and biomedical requirements. As an example, different materials and arrangements are analysed to find micropolar constitutive parameters that are comparable to bone parameters reported in the literature. This allows the GBR mesh to possess the mechanical performance that matches the adjacent bones.

Pulsatile flow of EMHD non-Newtonian nano-blood through an inclined tapered porous artery with combination of stenosis and aneurysm under body acceleration and slip effects

Research on blood flow in diseased arteries has become a priority in recent years. Stenosis/aneurysm is the most common arterial diseases. This work investigates the mixed effect of electric and magnetic fields on the unsteady, two-dimensional, laminar, non-Newtonian pulsatile flow of blood in an axisymmetrically inclined tapered porous artery with stenosis and aneurysm containing gold nanoparticles, subjected to body acceleration and slip effect. Thermal radiation and heat sources are also included. To immobilize the effect of the vessel wall, we use a radial coordinate transformation. The nonlinear partial differential equations governing the present problem, together with the prescribed boundary conditions, are solved using an explicit finite-difference scheme. The current analysis could help to understand the improvement in mass distribution and heat transfer in diseased blood vessels, which might be useful for drug delivery in the therapy of stenosis and aneurysm.

Computational heat transfer analysis of liquid food in a heat exchanger with varying stirrer settings in a nonclassical continuum framework

Rapid heat diffusion and stirring of food in channels and tanks for more desirable and safely regulated food delivery is of particular interest in food industry. In the present article, we investigate heat diffusion in a heat exchanger with varying stirrer settings in a nonclassical framework. Interestingly, the liquid food flows in all previous investigations are modeled using the classical continuous paradigm. Therefore, the purpose of this article is to offer a mathematical model that uses a nonclassical continuum framework to mimic the behavior of liquid food flows. This is influenced by the fact that during the stirring process, food particle microrotations affect the heat diffusion phenomenon, necessitating consideration of this in the modeling aspect of such studies. This non-Newtonian physical process is modeled based on the well-established theory of Cosserat continuum. The mathematical model presented here is taking into consideration a newly established constitutive relation by the author for the strain-based viscosity models in non-Newtonian mechanics. The applied constitutive relation incorporates the microrotational effects into the current rheology of rate-dependent viscosity model as a result of the kinematics at the macroscopic level. It offers a more practical method for examining the heat diffusion process in a heat exchanger. To the best of the author’s knowledge, this approach is first of its kind in analyzing heat diffusion process within non-Newtonian mechanics. The presented strain-based viscosity model is described in the form of PDEs along with prescribed boundary conditions. An in-house code is developed based on finite element method. The code is implemented using FreeFEM++. Liquid food flow in a heating process is analyzed numerically for heat transfer characteristics with varying Cosserat number and other physical aspects. Simulations are performed in a channel connecting a tank with different stirrer settings. Computed results are analyzed in a limiting case for accuracy and a strong agreement is achieved with the available solutions from the literature. Some interesting features of the liquid food flow and heat transfer are presented and discussed. It is observed that when liquid food particles rotate faster, the temperature difference between the walls also increases. The temperature difference between the top and bottom walls of the liquid food tank increases with faster speed of stirrers. Rise in the Cosserat number results in a dispersed vortices throughout the liquid food tank. The temperature differential between the liquid food tank’s top and bottom walls rises as a result of an increase in the microrotational motions of the food particles. Moreover, the microrotational motion of the liquid food particles rises with an increase in Cosserat number, enabling effective particle mixing.

Imposing different boundary conditions for thermal computational homogenization problems with FFT‐ and tensor‐train‐based Green's operator methods

AbstractTo compute the effective properties of random heterogeneous materials, a number of different boundary conditions are used to define the apparent properties on cells of finite size. Typically, depending on the specific boundary condition, different numerical methods are used. The article at hand provides a unified framework for Lippmann–Schwinger solvers in thermal conductivity and Dirichlet (prescribed temperature), Neumann (prescribed normal heat flux) as well as periodic boundary conditions. We focus on the explicit jump finite‐difference discretization and discuss different techniques for computing the discrete Green's operator. These include Fourier‐type methods, that is, based on discrete sine, cosine and Fourier transformations, as well as low‐rank tensor methods, that is, the tensor‐train approximation, whose computational prowess was demonstrated recently. We use the developed computational technology to investigate a number of interesting random materials and assess different microstructure‐generation techniques in terms of their compatibility with the prescribed boundary conditions.

On the convergence of the Willmore flow with Dirichlet boundary conditions

Very little is yet known regarding the Willmore flow of surfaces with Dirichlet boundary conditions. We consider surfaces with a rotational symmetry as initial data and prove a global existence and convergence result for solutions of the Willmore flow with initial data below an explicit, sharp energy threshold. Strikingly, this threshold depends on the prescribed boundary conditions — it can even be made to be 0. We show sharpness for some critical boundary data by constructing surfaces above this energy threshold so that the corresponding Willmore flow develops a singularity. Finally, a Li–Yau inequality for open curves in H2 is proved.

Ternary cobalt ferrite (CoFeO4)-silver (Ag)-titanium dioxide (TiO2) hybrid nanofluid hydromagnetic nonlinear radiative-convective flow from a rotating disk with viscous dissipation, non-Darcy and non-Fourier effects: Swirl coating simulation

Magnetic nanoparticles are increasingly being deployed in smart coating systems due to their exceptional functionalities and abilities to be tuned for specific environmental conditions. Inspired by the emergence of tri-hybrid magnetic nanofluids which utilize three distinct nanoparticles in a single base fluid coating, the present article examines analytically and computationally the swirl coating of magnetic ternary hybrid nanofluid from a rotating disk, as a simulation of spin coating deposition processes in materials manufacturing. Owing to high temperature fabrication conditions, thermal radiative heat transfer is also considered and a Rossleand flux model deployed. -- hybrid nanoparticles are considered with Ethylene Glycol-Water () base fluid. A filtration medium is also featured (porous medium) adjacent to the disk and the Darcy-Forchheimer model is deployed to simulate both bulk matrix porous drag encountered at lower Reynolds numbers and inertial quadratic drag generated at higher Reynolds numbers. Thermal relaxation of the coating nanofluid is additionally addressed and a non-Fourier Cattaneo-Christov model is therefore implemented in the heat conservation equation. Viscous dissipation is also included in the model. The governing conservation equations for mass, momenta (radial, tangential and axial) and energy with prescribed boundary conditions are rendered into coupled nonlinear ordinary differential boundary layer equations via suitable scaling variables and the Von Karman transformations. The derived reduced boundary value problem is then solved with a Runge-Kutta numerical scheme and shooting scheme in MATLAB. Validation of solutions is included with previous studies. Radial and azimuthal velocities, temperature, radial skin-friction, azimuthal skin friction and local Nusselt number are computed for a range of selected parameters. A comparative assessment of mono nanofluid Hybrid - nanofluid and tri-Hybrid -- nanofluid is conducted. This combination of hybrid nanoparticles has never been examined previously in the literature and constitutes the significant novelty of the present work. Both radial and tangential velocity are depleted with increasing applied magnetic field whereas temperature and thermal boundary layer thickness are increased.

A variational quantum algorithm-based numerical method for solving potential and Stokes flows

This paper presents a numerical method based on the variational quantum algorithm to solve potential and Stokes flow problems. In this method, the governing equations for potential and Stokes flows can be respectively written in the form of Laplace's equation and Stokes equations using velocity potential, stream function and vorticity formulations. Then the finite difference method and the generalized differential quadrature (GDQ) method are applied to discretize the governing equations. For the prescribed boundary conditions, the corresponding linear systems of equations can be obtained. These linear systems are solved by using the variational quantum linear solver (VQLS), which resolves the potential and Stokes flow problems equivalently. To the best of authors' knowledge, this is the first study that incorporates the GDQ method which is inherently a high-order discretization method with the VQLS algorithm. Since the GDQ method can utilize much fewer grid points than the finite difference method to approximate derivatives with a higher order of accuracy, the size of the input matrix for the VQLS algorithm can be smaller. In this way, the computational cost may be saved. The performance of the present method is comprehensively assessed by two representative examples, namely, the potential flow around a circular cylinder and Stokes flow in a lid-driven cavity. Numerical results validate the applicability and accuracy of the present VQLS-based method. Furthermore, its time complexity is evaluated by the heuristic scaling, which demonstrates that the present method scales efficiently in the number of qubits n and the precision ε. This work brings quantum computing to the field of computational fluid dynamics. By virtue of quantum advantage over classical methods, promising advances in solving large-scale fluid mechanics problems of engineering interest may be prompted.

The Upshot of Nonlinear Thermal Emission on a Conducting Jeffrey Nanofluid Flow over a Stretching Sheet: A Lie Group Approach

The upshot of nonlinear thermal radiation of steady state MHD heat transfer of Jeffrey nanofluid together with prescribed boundary conditions of interest was studied. Essential fluid properties, dimensionless switch parameters with the assistance of the Lie group method were used to transform the convenient partial differential equations that describe the present model into a system of ordinary differential equations. The generalized flow of the present model incorporates Jeffrey parameters and nonlinear thermal radiation. The Mathematical model is first renovated into ordinary differential equations by Lie symmetry group alteration. The renovated equations were solved numerically using a bvp4c MATLAB solver. The Jeffrey parameter serves as a stabilizer on the velocity of the fluid while thermal radiation parameter and heat source-sink parameter improves the flow temperature. Lewis number, chemical reaction parameter diminished the mass transfer flow. Equally skin friction, Sherwood number and Nusselt number were expanded.

Modeling of Love wave propagation in the diversified piezomagnetic layered structure under impulsive force due to a point source

Love wave propagation through smart materials is an interesting research area due to its possible applications in seismic acoustic wave (SAW) devices. Apart from existing research, we considered sinusoidal diversity in the piezomagnetic materials under the effect of an impulsive point source to elucidate the hidden characteristics of Love wave behavior. As a two-dimensional problem, the governing equations of motion are laid down for the proposed model and deduced by Fourier transformation. The frequency relation is derived by using prescribed boundary conditions and the well-known properties of Green’s function. The obtained frequency relation is deduced into the conventional form of the Love wave dispersion, validating the proposed model. The fundamental mode along with two higher-order modes of Love wave dispersion is investigated analytically to demonstrate the parametric effects on frequency curves. Analytical findings recommended that the functional gradient parameters, piezomagnetic constants, magnetic permeabilities, and amplify gradient parameters affected the phase velocity of the Love wave extensively. This finding suggested that it is important to take into account the suppressing effects of the parameters when manufacturing transducers and piezomagnetic sensors to achieve better performance. Moreover, the outcomes of this study may be considered to secure the high performance of the SAW devices.

Imposing slip conditions on curved boundaries for 3D incompressible flows with a very high-order accurate finite volume scheme on polygonal meshes

The conventional no-slip boundary condition does not always hold in several fluid flow applications and must be replaced with appropriate slip conditions according to the wall and fluid properties. However, not only slip boundary conditions are still a subject of discussion among fluid dynamicists, but also their numerical treatment is far from being well-developed, particularly in the context of very high-order accurate methods. The complexity of these conditions significantly increases when the boundary is not aligned with the chosen coordinate system and, even more challenging, when the fluid slips along a curved boundary. The present work proposes a simple and efficient numerical treatment of general slip boundary conditions on arbitrary curved boundaries for three-dimensional fluid flow problems governed by the incompressible Navier–Stokes equations. In that regard, two critical challenges arise: (i) achieving very high-order of convergence with arbitrary curved boundaries for the classical no-slip boundary conditions and (ii) extending the developed numerical techniques to impose general slip boundary conditions. The conventional treatment of curved boundaries relies on generating curved meshes to eliminate the geometrical mismatch between the physical and computational boundaries and achieve high-order of convergence. However, such an approach requires sophisticated meshing algorithms, cumbersome quadrature rules on curved elements, and complex non-linear transformations. In contrast, the reconstruction for off-site data (ROD) method handles arbitrary curved boundaries approximated with linear piecewise elements while employing polynomial reconstructions with specific linear constraints to fulfil the prescribed boundary conditions. For that purpose, the general slip boundary conditions are reformulated on a local orthonormal basis to allow a straightforward application of the ROD method with scalar boundary conditions. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed solely on the polygonal mesh elements. Several benchmark test cases of fluid flow problems in non-trivial three-dimensional curved domains are addressed and confirm that the proposed method effectively achieves up to the eighth-order of convergence.

Heat and mass transfer for MHD nanofluid flow on a porous stretching sheet with prescribed boundary conditions

A theoretical study is conducted in order to scrutinize the thermodynamic first law of the MHD nanofluid flow with an inclined magnetic field, radiation, heat source/sink, viscous dissipation, Joule heating, concentration power-law exponent, and the chemical reaction on a porous stretching surface immersed within a permeable Darcian medium. Cobalt ferrite nanoparticles (CoFe2O4) have been combined with pure water to form a nanofluid called CoFe2O4/H2O. The controlling mathematical equations for MHD nanofluid flow are transformed through similarity transformation into non-dimensional equations. The exact solutions for the energy and mass transfer equations are provided in terms of confluent hypergeometric function. The effects of controlling parameters on the velocity, temperature, and concentration profiles are discussed and illustrated with figures and tables. According to the results, increasing the concentration power-law exponent yields a rise in the Sherwood number (PSC) magnitude and the wall concentration (PMF). Furthermore, the 3-D plots showed that the skin friction coefficient is directly related to the Hartmann number, suction parameter, and nanoparticle volume fraction parameter.

HESS Opinions: Are soils overrated in hydrology?

Abstract. Traditional hydrological theories are based on the assumption that soil is key in determining water's fate in the hydrological cycle. According to these theories, soil hydraulic properties determine water movement in both saturated and unsaturated zones, described by matrix flow formulas such as the Darcy–Richards equations. They also determine plant-available moisture and thereby control transpiration. Here we argue that these theories are founded on a wrong assumption. Instead, we advocate the reverse: the terrestrial ecosystem manipulates the soil to satisfy specific water management strategies, which are primarily controlled by the ecosystem's reaction to climatic drivers and by prescribed boundary conditions such as topography and lithology. According to this assumption, soil hydraulic properties are an effect rather than a cause of water movement. We further argue that the integrated hydrological behaviour of an ecosystem can be inferred from considerations about ecosystem survival and growth without relying on internal-process descriptions. An important and favourable consequence of this climate- and ecosystem-driven approach is that it provides a physical justification for catchment models that do not rely on soil information and on the complexity associated with the description of soil water dynamics. Another consequence is that modelling water movement in the soil, if required, can benefit from the constraints that are imposed by the embedding ecosystem. Here we illustrate our ecosystem perspective of hydrological processes and the arguments that support it. We suggest that advancing our understanding of ecosystem water management strategies is key to building more realistic hydrological theories and catchment models that are predictive in the context of environmental change.

Heat transfer through a higher grade Forchheimer porous CuO–H2O-nano-medium confined between non-isothermal moving plates

In this study, the aim is to analyze numerically heat transfer through a higher grade Forchheimer porous CuO–H2O-nano-medium in an unsteady magneto-hydrodynamic (MHD) porous flow. The nano-medium is confined between two non-isothermal moving infinite parallel plates. The flow governing equations are developed through variational calculus approach and are resulted in partial differential equations. The temperature of both the plates are taken to be variable. The governing flow PDEs along with the prescribed boundary conditions are solved using finite difference approach for time discretization and finite element approach for discretizing the spatial coordinate. The algorithm used is described here in detail for the numerical simulations and calculations of results. The proposed algorithm is implemented through MATLAB and simulations are performed for parameters of interest which includes: Nusselt number, skin friction coefficients, Forchheimer constant, volume fraction, Prandtl number, Grashhoff number, Hartmann number, and convection and porosity parameters. Effective implementation of the proposed algorithm is demonstrated through plots of time profiles of non-dimensionalized thermodynamic velocity and temperature against these parameters. It is observed that the larger the porosity of the nanomedium leads to the larger skin friction at both the plates. The plate's temperature gets increased with the increase in the Forchhiemer number. However, the hydrodynamic velocities get smaller as the porosity gets elevated. Forchheimer constant and skin friction coefficients are related with direct proportionality. Whereas the skin friction coefficient is observed behaving monotone in relation to the Hartman constant. The Nusselt number at both the plates get increase in its value as the nanoparticles concentration increases. By increasing the velocity ratio between the two moving plates shows opposite trend. However, the Nusselt number decreases if the porosity of the nano-fluidic medium is increased. It is pertinent to mention that the present modeling approach can be used to analyze the Forchheimer medium flows involving catalytic converters and gas turbine dynamics.

Calibration of patient-specific boundary conditions for coupled CFD models of the aorta derived from 4D Flow-MRI.

Introduction: Patient-specific computational fluid dynamics (CFD) models permit analysis of complex intra-aortic hemodynamics in patients with aortic dissection (AD), where vessel morphology and disease severity are highly individualized. The simulated blood flow regime within these models is sensitive to the prescribed boundary conditions (BCs), so accurate BC selection is fundamental to achieve clinically relevant results. Methods: This study presents a novel reduced-order computational framework for the iterative flow-based calibration of 3-Element Windkessel Model (3EWM) parameters to generate patient-specific BCs. These parameters were calibrated using time-resolved flow information derived from retrospective four-dimensional flow magnetic resonance imaging (4D Flow-MRI). For a healthy and dissected case, blood flow was then investigated numerically in a fully coupled zero dimensional-three dimensional (0D-3D) numerical framework, where the vessel geometries were reconstructed from medical images. Calibration of the 3EWM parameters was automated and required ~3.5 min per branch. Results: With prescription of the calibrated BCs, the computed near-wall hemodynamics (time-averaged wall shear stress, oscillatory shear index) and perfusion distribution were consistent with clinical measurements and previous literature, yielding physiologically relevant results. BC calibration was particularly important in the AD case, where the complex flow regime was captured only after BC calibration. Discussion: This calibration methodology can therefore be applied in clinical cases where branch flow rates are known, for example, via 4D Flow-MRI or ultrasound, to generate patient-specific BCs for CFD models. It is then possible to elucidate, on a case-by-case basis, the highly individualized hemodynamics which occur due to geometric variations in aortic pathology high spatiotemporal resolution through CFD.

Mathematical modelling of nonlinear pressure drops in arbitrarily shaped port utilizing dual boundary element method

A mathematical model is developed to analyze the wave-trapping characteristics of different types of porous and non-porous barriers in arbitrarily shaped domains utilizing the iterative Dual Boundary Element Method (DBEM) under the resonance conditions. In this analysis, Vertical, Wavy, Inverted V-shaped, and Inverted L-shaped barriers are utilized to investigate the effect of incident waves over the boundary of the harbor. The harbor is divided into an interconnected domain with variable bathymetry, and the Helmholtz equation is solved in each subdomain utilizing the prescribed boundary conditions. Further, the semi-empirical discharge equation with quadratic pressure drop is considered on the porous barrier to incorporate the effect of incident wave amplitude on the energy loss. The present numerical scheme is validated with existing studies and verified with convergence analysis. Further, the mathematical model is implemented on the Visakhapatnam port at four key locations with improved performance of different types of barriers. Moreover, the influence of the partial reflection coefficient is also investigated by considering the various combination of partially reflecting boundaries of the port. The wave trapping or absorbing characteristics of porous barriers proves an efficient tool to reduce the wave resonance inside the port.

Visualization of thermo-magnetic natural convective heat flow in a square enclosure partially filled with a porous medium using bejan heatlines and Hooman energy flux vectors: Hybrid fuel cell simulation

The aim of the article is to study the magneto-convective laminar incompressible flow within a differentially heated square enclosure partly filled with porous medium saturated with ionized helium gas in the presence of a transverse magnetic flux, as a model for emerging fuel cell designs. A Darcy model is utilized for porous medium bulk drag effects. Visualization of heat line and energy flux vectors is computed via Bejan's heat line approach and the Hooman energy flux vector method. The horizontal (i.e., bottom and top) wall boundaries are considered adiabatic and impermeable, while the side walls (cold and hot walls) are maintained at different but constant temperatures. The nonlinear coupled conservation equations under prescribed boundary conditions are solved with a finite difference-based vorticity-stream function approach. Appropriate convergence criteria factors are deployed. Furthermore, validation with earlier studies for the purely fluid, non-magnetic case i. e. in the absence of porous medium (Da) and Hartmann number (Ha) effects, is included. A parametric examination of the impact of Rayleigh number (Ra), Darcy number (Da) and Hartmann number (Ha) on temperature contours, heat lines, energy flux vectors and streamline patterns for ionized helium gas (Prandtl number, Pr = 0.71) is conducted. An increment in Hartmann magnetic number decreases the magnitudes of heat lines, suppresses heat transmission in the enclosure and curtails flow circulation. Enhancing the magnetic parameter and Rayleigh number however boosts the average heat transfer rate. Average Nusselt number at the left hot wall is elevated with increasing permeability of the porous medium and Rayleigh number. Mid-section velocities are enhanced in the porous layer but depleted in the non-porous layer of the enclosure with greater Rayleigh number. The simulations find provide some insight into applications in emerging hybrid electromagnetic fuel cells enabling better visualization of the thermal/fluid characteristics via the novel heat flow visualization heat-function and energy flux vectors analogy.

Conductivity reconstruction from power density data in limited view

In acousto-electric tomography, the objective is to extract information about the interior electrical conductivity in a physical body from knowledge of the interior power density data generated from prescribed boundary conditions for the governing elliptic partial differential equation. In this note, we consider the problem when the controlled boundary conditions are applied only on a small subset of the full boundary. We demonstrate using the unique continuation principle that the Runge approximation property is valid also for this special case of limited view data. As a consequence, we guarantee the existence of finitely many boundary conditions such that the corresponding solutions locally satisfy a non-vanishing gradient condition. This condition is essential for conductivity reconstruction from power density data. In addition, we adapt an existing reconstruction method intended for the full data situation to our setting. We implement the method numerically and investigate the opportunities and shortcomings when reconstructing from two fixed boundary conditions.

Impact of unsteadiness on the non-Newtonian flow of blood in a vascular tube with stenosis and aneurysm: analytical solution

A theoretical analysis is conducted to explore the time-dependent problem of blood flow streaming in a tapered vascular tube having post-stenotic dilatation followed by symmetric arterial stenosis. Blood is considered a homogeneous incompressible fluid whose rheological behavior is estimated by Herschel–Bulkley fluid model. By exploiting the prescribed boundary conditions, unique expressions of velocity, flow rate, plug core radius and resistance to the flow have been made by handling the complexity of the governing equations with the standard perturbation technique. The steady-state component of the pressure gradient originating due to the functioning of the heart is computed numerically. The effects of various parameters involved in the problem in accordance with the stenosis and aneurysm on the different flow variables are elaborated with the help of graphs. It has been investigated that the rheological parameters have considerable effects on the dynamic of the flow throughout the constricted region of the vascular tube.

A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution

In this paper, a new method based on an accurate analytical series solution for free vibration of triangular isotropic and orthotropic plates is presented. The proposed solution is expressed in terms of undetermined arbitrary coefficients, which are exactly satisfied by the governing differential equation in free vibration. The approach used is based on an innovative extension of the superposition method through the application of a modified system of trigonometric functions. The boundary conditions for bending displacements and bending rotations on the sides of the triangular plate led to an infinite system of linear algebraic equations in terms of the undetermined coefficients. Following this development, the paper then presents an algorithm to solve the boundary value problem for isotropic and orthotropic triangular plates for any kinematic boundary conditions. Of course, the boundary conditions with zero displacements and zero rotations on all sides correspond to the case when the plate is fully clamped all around. The convergence of the proposed method is examined by numerical simulation applying stringent accuracy requirements to fulfill the prescribed boundary conditions. Some of the computed numerical results are compared with published results and finally, the paper draws significant conclusions.