Navier Equations Research Articles

Transient hydrodynamic, heat and mass transfer in a salinity gradient solar pond: A numerical study

The impoverishment of our planet in non-renewable energies has incited researchers to design salinity gradient solar ponds to collect and store solar energy at a lower cost. It is in this context that the present research work lies to focus on the numerical study of the transient hydrodynamic, heat and mass transfer in a salinity gradient solar pond. The problem is tackled using the dimensionless governing equations of Navier–Stokes, thermal energy and mass transfer, which are solved numerically by finite-volume method to provide the temperature, concentration and velocity fields in transient regime. The pond is filled with salty water of various salinities to form three zones of salty water: Upper Convective Zone (UCZ), Non-Convective Zone (NCZ) and Lower Convective Zone (LCZ). To prevent convective movements induced by the internal heating of salty water due to solar radiation absorption, a salinity gradient is used in the solar pond. Representative results illustrating the influence of internal Rayleigh number on the thermal performance of the pond and the effect of the aspect ratio on the distribution of temperature and velocity fields in the salinity gradient solar pond (SGSP) are discussed. In addition, results for the transient average temperature of UCZ and LCZ are presented and discussed for various parametric conditions.

Nonlocal elasticity defined by Eringen’s integral model: Introduction of a boundary layer method

In this paper we consider a nonlocal elasticity theory defined by Eringen’s integral model and introduce, for the first time, a boundary layer method by presenting the exponential basis functions (EBFs) for such a class of problems. The EBFs, playing the role of the fundamental solutions, are found so that they satisfy the governing equations on an unbounded domain. Some insight to the theory is given by showing that the EBFs satisfying the Navier equations in the classical elasticity theory also satisfy the governing equations in the nonlocal theory. Some additional EBFs are particularly obtained for the nonlocal theory. In order to use the EBFs on bounded domains, the effects of the boundary conditions are taken into account by truncating the kernel/attenuation function in the constitutive equations. This leads to some residuals in the governing equations which appear near the boundaries. A weighted residual approach is employed to minimize the residuals near the boundaries. The method presented in this paper has much in common with Trefftz methods especially when the influence area of the kernel function is much smaller than the main computational domain. Several one/two dimensional problems are solved to demonstrate the way in which the EBFs can be used through the proposed boundary layer method.

Elastic wave propagation in unsaturated porous media using hybrid‐Trefftz stress elements

SUMMARYThe stress model of the hybrid‐Trefftz finite element is formulated for the analysis of elastodynamic problems defined on unsaturated porous media. The supporting mathematical model is the theory of mixtures with interfaces and considers the full coupling between the solid, fluid and gas phases, including the effect of seepage acceleration. Hybrid‐Trefftz stress elements use the free‐field regular solutions of the homogeneous Navier (or Beltrami) equation to construct the approximation of the generalized stresses in the domain of the element. The influence of non‐homogeneous terms in the Navier equation is modelled using solutions of the corresponding static problem. The resulting elements are highly convergent under p‐refinement and robust to both low and high excitation frequencies, as the trial functions embody relevant physical information on the modelled phenomenon. Copyright © 2013 John Wiley & Sons, Ltd.

Mathematical modeling of the thermal discharge under various operational capacities of thermal power plants

This paper presents a mathematical model of the thermal discharge under various operational capacities of thermal power plants, which is solved by the equations of Navier - Stokes and temperature for an incompressible fluid in a stratified medium based on the method of splitting by physical parameters that can be discretize by the control volume method. In the first step it is assumed that the transfer of momentum carried out only by convection and diffusion. Intermediate velocity field is solved by 5-step Runge - Kutta method. At the second stage, the pressure field is solved based on the found intermediate velocity field. The algorithm is parallelized on high-performance systems. The obtained numerical results of three-dimensional stratified turbulent flow reveals to approximate qualitatively and quantitatively the basic laws of hydrothermal processes occurring in the aquatic environment.

Effect of Induced Vibration on the Blood Flow Properties in a Mechanical Aortic Valve

The effect of induced vibration on velocity distribution for the blood flow in the bileaflet mechanical heart valves conveying blood was investigated in this study. The bileaflet valve was simulated as an orifice. The induced vibration is due to the pulsed blood flow in the artery. Results presented in this study were performed using CFD FLUENT software. This analysis is based on the non-linear numerical solution by using a finite-element method, for the system of governing partial differential equations (continuity, momentum) of Navier - Stokes equation of blood flow through the orifice. It has been found that as the flow through the orifice increased, the vibration at the orifice inlet increased. For steady state conditions, at Reynolds number 50 the recorded frequency was 20Hz. When the Reynolds number increased to 100 due to the increase in the blood flow the recorded frequency increased to 30Hz. The increase in frequency may result in damaging the blood red cells and platelets which subsequently results in increasing the blood clogging downstream of the orifice.

Новый подход к исследованию механических свойств многослойного графена с помощью метода конечных элементов

Saratov State University, 83, Astrahanskaya str., 410012, Saratov, Russia, oeglukhova@yandex.ru, dzero@pisem.net,kolesnikova.88@mail.ru, vshunaev@list.ruA new approach to investigate the mechanical properties of multilayer graphene was suggested. The method is based on the ideathatthevanderWaalsinteractionbetweenthegraphenesheetscanbesimulatedbyafictitiouslayerofcontinuum.Thestress-strainstate of multilayer graphene is described by stationary equations of Navier–Lame. This approach has been successfully tested ongraphene deflection. The graphene layers were considered as linear-elastic material. For each part of the curve that approximatesthe dependence of the graphene deflection on the applied force, corresponding elastic constants of graphene layers were found.Key words: multilayer graphene, finite element method, deflection, modulus of elasticity, van der Waals interactions.

Free vibration of simply supported rectangular plates on Pasternak foundation: An exact and three-dimensional solution

This paper deals with exact solution for free vibration analysis of simply supported rectangular plates on elastic foundation. The solution is on the basis of three dimensional elasticity theory. The foundation is described by the Pasternak (two-parameter) model. First, the Navier equations of motion are replaced by three decoupled equations in terms of displacement components. Then, these equations are solved in a semi-inverse method. The obtained displacement field satisfies all the boundary conditions of the problem in a point wise manner. The solution is in the form of a double Fourier sine series. Then free-vibration characteristics of rectangular plates resting on elastic foundations for different thickness/span ratios and foundation parameters are studied. The numerical results are compared with the available results in the literature. Important parameters on the accuracy of plate theories and freevibration characteristics of rectangular plates resting on elastic foundations are discussed.

IPMC mechanoelectrical transduction: its scalability and optimization

Ionic polymer–metal composite (IPMC) mechanoelectrical transduction is described with a system of partial differential equations—the Poisson equation, the Nernst–Planck equation, and the Navier equations for the displacements. Modeling of this system of four equations is challenging due to the differences in the physical fields—namely, one physical field can be very smooth while others have steeper gradients. When using the conventional finite element method (FEM), the problem size in terms of number of degrees of freedom can be very large. Additionally, it is challenging to find an optimal mesh due to the fact that the physical fields are time dependent. Last but not least, due to the coupled nature of the system, it is necessary to have a way to estimate the error to ensure the desired accuracy of the solutions. In this work, we propose a novel hp-FEM modeling method for solving the system of equations. hp-FEM is a modern version of the FEM that is capable of exponential convergence. It is also demonstrated how the multi-meshing reduces the problem size while maintaining the prescribed error limit. The solution domain that describes IPMC can be scaled without a significant increase in the number of degrees of freedom and solution time. Additionally, PID controller based time step optimization is incorporated. The model is implemented in Hermes, which is a free hp-FEM solver.

Thermal residual stresses in in-plane functionally graded clamped hollow circular plates subjected to an edge heat flux

This study carries out the thermal residual stress analyses of functionally graded clamped hollow circular plates for in-plane constant inner and outer edge heat fluxes. The material properties of the functionally graded plates were assumed to vary with a power law along an in-plane direction not through the plate thickness direction. The transient heat conduction and Navier equations describing the two-dimensional thermoelastic problem were discretized using finite difference method, and the set of linear equations were solved using the pseudo singular value method. In order to determine the effect of the plate material properties on the thermal deformation and stress states the circular plates were designed in the way that their material compositions can vary from a pure ceramic (C) outer edge to a pure metal (M) inner edge and vice versa, such as ceramic-to-metal or metal-to-ceramic circular plates. The compositional gradient and direction affected considerably both in-plane temperature levels and heat transfer period whereas similar temperature distributions existed. The displacement components exhibited similar symmetrical distributions and were at lower levels in the metal-rich compositions. The normal strain components had similar distributions to those of the displacement components and were critical around both the inner and outer edges. A metal-rich composition resulted in lower normal strain levels. The equivalent strain distributions were always critical around the heat flux edge and became highest for a ceramic-rich composition. The normal stress components and equivalent stresses were critical in the ceramic-rich regions and the equivalent stress exhibited a sudden increase near the pure ceramic-edge subjected to the heat flux. The lower stress levels were also observed for a metal-rich composition variation. The ceramic material, a good thermal insulator, makes a metal-to-ceramic and a ceramic-to-metal material system suitable for outer and inner edge heat fluxes, respectively in practice. The metal-rich material compositions (with [Formula: see text] and 10.0) can prevent the local cracks in the ceramic-rich regions induced by high heat gradients.

Using equilibrium equation constraints to obtain precise lateral displacement estimates in ultrasound imaging

Ultrasound elasticity imaging, whether based on radiation force, quasistatic deformation, or other means, depends upon using ultrasound to measure tissue motion. Ultrasound image data with standard beamforming can provide very precise measurements of soft tissue displacement in the axial direction, i.e., in the direction of the ultrasound beam. Lateral (and elevational) displacement estimates are relatively noisy. The authors describe a new processing method designed to estimate a precise and accurate 2D full displacement vector field from accurate measurements of a single component, and a noisy measurement of a second component. The proposed variational approach finds the displacement field that best fits the data, but that satisfies Navier's equation locally. The equilibrium equation is automatically relaxed along internal interfaces in the material, which need not be identified beforehand. In this way, the method accommodates piecewise constant material property distributions without knowing in advance where the properties change. The iterative implementation for plane stress incompressible elasticity converges in 3–10 iterations. [Authors gratefully acknowledge funding from NSF and NIH (NSF Grant No. 50201109; NIH NCI-R01CA140271).]

On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime

According to the Helmholtz decomposition, the irrotational parts of the momentum balance equations of the incompressible Navier–Stokes equations are balanced by the pressure gradient. Unfortunately, nearly all mixed methods for incompressible flows violate this fundamental property, resulting in the well-known numerical instability of poor mass conservation. The origin of this problem is the lack of L2-orthogonality between discretely divergence-free velocities and irrotational vector fields. Therefore, a new variational crime for the nonconforming Crouzeix–Raviart element is proposed, where divergence-free, lowest-order Raviart–Thomas velocity reconstructions reestablish L2-orthogonality. This approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence-free flow solvers. In the Stokes case, optimal a priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.

To solve the inverse Cauchy problem in linear elasticity by a novel Lie-group integrator

In this paper, we propose a simple, iteration free and easy-to-implement numerical algorithm for the solution of inverse Cauchy problem in linear or nonlinear elasticity. The bottom of a finite rectangular plate is imposed by overspecified boundary data, and we seek unknown data on the top side. A spring-damping transform method (SDTM) is introduced to the Navier equations, such that after a discretization by the differential quadrature method, we can apply a novel Lie-group integrator, namely the mixed group-preserving scheme (MGPS), to solve them as an initial value problem. Several numerical examples including nonlinear ones are examined to show that the MGPS can overcome the ill-posed behaviour of the inverse Cauchy problem in elasticity, which has good efficiency and stability against the noisy disturbance, even with an intensity large up to and .

The factorization method for inverse elastic scattering from periodic structures

This paper is concerned with the inverse problem of scattering of time-harmonic elastic waves from rigid periodic structures. We establish the factorization method to identify an unknown diffraction grating profile (periodic surface) from knowledge of the scattered compressional or shear waves measured on a line above the periodic surface. Near-field operators are factorized by selecting appropriate incident waves derived from quasi-periodic half-space Green's tensor to the Navier equation. The factorization method gives rise to a uniqueness result for the inverse scattering problem by utilizing only the compressional or shear components of the scattered field corresponding to all quasi-periodic incident plane waves with a common phase-shift. A number of computational examples are provided to show the accuracy of the inversion algorithms, with emphasis placed on comparing reconstructions from the scattered near field and those from its compressional and shear components.

Nonlocal Constrained Value Problems for a Linear Peridynamic Navier Equation

In this paper, we carry out further mathematical studies of nonlocal constrained value problems for a peridynamic Navier equation derived from linear state-based peridynamic models. Given the nonlocal interactions effected in the model, constraints on the solution over a volume of nonzero measure are natural conditions to impose. We generalize previous well-posedness results that were formulated for very special kernels of nonlocal interactions. We also give a more rigorous treatment to the convergence of solutions to nonlocal peridynamic models to the solution of the conventional Navier equation of linear elasticity as the horizon parameter goes to zero. The results are valid for arbitrary Poisson ratio, which is a characteristic of the state-based peridynamic model.

An alternative local collocation strategy for high-convergence meshless PDE solutions, using radial basis functions

This work proposes an alternative decomposition for local scalable meshless RBF collocation. The proposed method operates on a dataset of scattered nodes that are placed within the solution domain and on the solution boundary, forming a small RBF collocation system around each internal node. Unlike other meshless local RBF formulations that are based on a generalised finite difference (RBF-FD) principle, in the proposed “finite collocation” method the solution of the PDE is driven entirely by collocation of PDE governing and boundary operators within the local systems. A sparse global collocation system is obtained not by enforcing the PDE governing operator, but by assembling the value of the field variable in terms of the field value at neighbouring nodes. In analogy to full-domain RBF collocation systems, communication between stencils occurs only over the stencil periphery, allowing the PDE governing operator to be collocated in an uninterrupted manner within the stencil interior.The local collocation of the PDE governing operator allows the method to operate on centred stencils in the presence of strong convective fields; the reconstruction weights assigned to nodes in the stencils being automatically adjusted to represent the flow of information as dictated by the problem physics. This “implicit upwinding” effect mitigates the need for ad-hoc upwinding stencils in convective dominant problems. Boundary conditions are also enforced within the local collocation systems, allowing arbitrary boundary operators to be imposed naturally within the solution construction.The performance of the method is assessed using a large number of numerical examples with two steady PDEs; the convection–diffusion equation, and the Lamé–Navier equations for linear elasticity. The method exhibits high-order convergence in each case tested (greater than sixth order), and the use of centred stencils is demonstrated for convective-dominant problems. In the case of linear elasticity, the stress fields are reproduced to the same degree of accuracy as the displacement field, and exhibit the same order of convergence. The method is also highly stable towards variations in basis function flatness, demonstrating significantly improved stability in comparison to finite-difference type RBF collocation methods.

Spline-Based Hybrid Image Registration using Landmark and Intensity Information based on Matrix-Valued Non-radial Basis Functions

We introduce a new approach for spline-based elastic image registration using both point landmarks and intensity information. With this approach, both types of information as well as a regularization based on the Navier equation are directly integrated in a single energy minimizing functional. For this functional we have derived an analytic solution, which is based on matrix-valued non-radial basis functions. With our approach the full 3D intensity information is exploited, i.e., all voxels are considered and subsampling using a grid is not required. A special case of our hybrid approach is obtained by disregarding the landmark information, which results in a pure intensity-based elastic registration approach. We have successfully applied our approach to 3D synthetic images, 2D MR images of the human brain, 2D gel electrophoresis images, and 3D CT lung images.

Block Krylov Methods to Solve Adjoint Problems in Aerodynamic Design Optimization

A new resolution method for solving linear systems resulting from the discrete adjoint approach in aerodynamic shape optimization with a single objective and multiple constraint functions is presented. The steady flow is governed by the three-dimensional compressible Favre-averaged Navier–Stokes equations combined with a two-equation turbulence model. Block flexible variants of the generalized minimal residual method are Krylov methods designed for the solution of linear systems with multiple right-hand sides allowing variable preconditioning. Then, they can be applied to the discrete adjoint equations of Favre-averaged Navier–Stokes equations for all objective and constraint functions as a right-hand side. The paper focuses on the development of a new algorithm combining the three following features: flexibility, multiple right-hand-side concurrent processing, and spectral deflation. This latter property enables to recycle approximate spectral information from one restart to the next one in order to avoid generalized minimal residual stagnation when short restart recurrences are performed. This novel block method is named block flexible generalized minimal residual with deflated restarting. It is evaluated on two aerodynamic configurations showing its capabilities to reduce computational times over standard Krylov methods.

Thermal residual stresses in adhesively bonded in-plane functionally graded clamped circular hollow plates

This study investigates the effects of in-plane compositional gradient exponent and direction on the thermal residual stress and deformations in adhesively bonded functionally graded clamped circular plates. The material composition was assumed to vary with a power law along an in-plane direction not through the plate thickness direction. The transient heat conduction and Navier equations in polar coordinates describing the two-dimensional thermo-elastic problem were discretized using finite-difference method, and the set of linear equations were solved using the pseudo-singular-value method. The material composition direction is designed as Ceramic-Metal (CM)–CM, CM–Metal-Ceramic (MC), MC–CM, and MC–MC for the inner and outer plates. The temperature decreased radially along the plates, but exhibited a sharp decrease across the adhesive layer. The compositional gradient exponent and direction affected evidently temperature levels and heat transfer period. The compressive radial and shear strains are more effective on the deformation in the adhesive layer and the plate regions near the plate–adhesive interfaces. The adhesive layer is subjected to considerable shear deformations. The equivalent strain and stresses are very low in a large region of the plates but exhibit sharp peaks on the plate regions near the plate–adhesive interfaces, and decrease towards the adhesive interfaces. These stress and strain peaks in the plates and adhesive layer are affected by the compositional gradient and direction. For an outer edge flux, the largest equivalent strain and stresses are observed in the CM–MC joint but the lowest levels occur in the MC–CM or secondly CM–CM joint. In addition, an inner edge flux results in the lowest and highest peak strains and stresses in the MC–CM and CM–MC joints, respectively. The MC–MC and CM–CM joints result in lower temperature, stress and strain levels around the adhesive layer and along the adhesive interfaces for outer and inner edge fluxes, respectively.

Intensity-based hierarchical elastic registration using approximating splines

We introduce a new hierarchical approach for elastic medical image registration using approximating splines. In order to obtain the dense deformation field, we employ Gaussian elastic body splines (GEBS) that incorporate anisotropic landmark errors and rotation information. Since the GEBS approach is based on a physical model in form of analytical solutions of the Navier equation, it can very well cope with the local as well as global deformations present in the images by varying the standard deviation of the Gaussian forces. The proposed GEBS approximating model is integrated into the elastic hierarchical image registration framework, which decomposes a nonrigid registration problem into numerous local rigid transformations. The approximating GEBS registration scheme incorporates anisotropic landmark errors as well as rotation information. The anisotropic landmark localization uncertainties can be estimated directly from the image data, and in this case, they represent the minimal stochastic localization error, i.e., the Cramér-Rao bound. The rotation information of each landmark obtained from the hierarchical procedure is transposed in an additional angular landmark, doubling the number of landmarks in the GEBS model. The modified hierarchical registration using the approximating GEBS model is applied to register 161 image pairs from a digital mammogram database. The obtained results are very encouraging, and the proposed approach significantly improved all registrations comparing the mean-square error in relation to approximating TPS with the rotation information. On artificially deformed breast images, the newly proposed method performed better than the state-of-the-art registration algorithm introduced by Rueckert et al. (IEEE Trans Med Imaging 18:712-721, 1999). The average error per breast tissue pixel was less than 2.23 pixels compared to 2.46 pixels for Rueckert's method. The proposed hierarchical elastic image registration approach incorporates the GEBS approximation scheme extended with anisotropic landmark localization uncertainties as well as rotation information. Our experimental results show that the proposed scheme improved the registration result significantly.

Surface Waves on a Half-Space Due to a Time-Harmonic Loading on an Inertial Strip Foundation

An analytical approach has been applied to obtain the solution of Navier's equation for a homogeneous, isotropic half-space under an inertial foundation subjected to a time-harmonic loading. Displacement potentials were used to change the Navier's equation to a system of wave-type equations. Calculus of variation was employed to demonstrate the contribution of the foundation's inertial effects as boundary conditions. Use of the Fourier transformation method for the system of Poisson-type equations and applying the boundary conditions yielded the transformed surface displacement field. Direct contour integration has been employed to achieve the surface waves. In order to clarify the foundation's inertial effects, related coefficients were defined and a parametric study was conducted. Final results revealed that increasing the mass per unit length of the foundation or the frequency of the applied harmonic load intensifies the inertial effect factors. On the other hand, an increase in strip width or Poisson's ratio would imply reduction in the inertial effect factors.