Navier Equations Research Articles

Numerical Investigation of Sediment Transport of Sandy Beaches by a Tsunami-Like Solitary Wave Based on Navier–Stokes Equations

To improve our current understanding of tsunami-like solitary waves interacting with sandy beach, a nonlinear three-dimensional numerical model based on the computational fluid dynamics (CFD) tool OpenFOAM® is first self-developed to better describe the wave propagation, sediment transport, and the morphological responses of seabed during wave runup and drawdown. The finite volume method (FVM) is employed to discretize the governing equations of Navier–Stokes equations, combining with an improved volume of fluid (VOF) method to track the free surface and a k–ε model to resolve the turbulence. The computational capability of the hydrodynamics and the sediment transport module is well calibrated by laboratory data from different published references. The results verify that the present numerical model can satisfactorily reproduce the flow characteristics, and sediment transport processes under a tsunami-like solitary wave. The water-sediment transport module is then applied to investigate the effects of prominent factors, such as wave height, water depth, and beach slope, in affecting the beach profile change. Finally, a dimensionless empirical equation is proposed to describe the transport volume of onshore sediment based on simulation results, and some proper parameters are recommended through the regression. The results can be significantly helpful to evaluate the process of transported sediment by a tsunami event.

Wave dispersion characteristics of heterogeneous nanoscale beams via a novel porosity-based homogenization scheme

The important effect of porosity on the mechanical behaviors of a continuum, makes it necessary to be accounted for while analyzing the structure. Motivated by this fact, a new porosity-dependent homogenization scheme is presented in this article to investigate the wave propagation responses of functionally graded (FG) nanobeams by considering the coupling effects between density and Young’s moduli in porous materials. In the introduced homogenization method, which is a modified form of the power-law model, dependency of effective Young’s modulus to the mass density is covered. Based on the Hamilton principle, the Navier equations are developed using the Euler-Bernoulli beam model. Thereafter, the constitutive equations are obtained employing the nonlocal elasticity theory of Eringen. Next, the governing equations are solved in order to reach the wave frequency. Once the validity of presented methodology is proved, a set of parametric studies are adopted to emphasize the role of each variant on the wave dispersion behaviors of porous FG nanobeams.

Vibration analysis of porous metal foam shells rested on an elastic substrate

In this article, the vibration problem of an embedded cylindrical shell consisted of porous metal foam is solved via an analytical method with respect to the influences of various porosity distributions. Three types of porosity distribution across the thickness are covered, namely, uniform, symmetric, and asymmetric. The strain–displacement relations of the shell are assumed to be derived on the basis of the first-order shear deformation shell theory. Then, the achieved relations will be incorporated with the Hamilton’s principle in order to reach the Navier equations of the cylindrical shell. Next, the well-known Galerkin’s method is utilized to calculate the natural frequencies of the system. The influences of both simply supported and clamped boundary conditions are included. In order to show the accuracy of the presented method, the results of the present research are compared with those reported by former published papers. The reported results show that an increase in the porosity coefficient can decrease the frequency of the shell. Also, the stiffness of the system can be lesser decreased while symmetric porosity distribution is chosen.

The Complementary Functions Method Solution to the Functionally Graded Polar Orthotropic Rotating Hyperbolic Disks with Both Radially and Circumferentially Aligned Fibers

In the present study, efforts have been made to numerically evaluate elastic displacements and stresses in a convergent or divergent hyperbolic disk subjected to a centrifugal force of constant circular velocity. The disk material is assumed to be continuously radially functionally graded (FG) with two orthotropic materials based on the simplest Voigt rule with a power of volume fraction of two constituents. The fibers are assumed to be aligned along either radial (RR) or circumferential (CR) directions. Having been a second order differential equation with variable coefficients, the governing equation so-called Navier equation is first derived and then put in the form of two differential equations of first order. These two ordinary differential equation set is originally solved based on the initial value problem (IVP) by employing the Complementary Functions Method (CFM). The numerical results are verified with the corresponding benchmark results for uniform thickness FG polar orthotropic disks. The radial variation of the elastic fields in a hyperbolic disk is investigated for several boundary conditions, disk profile parameters, and the gradient parameter for both the radially and circumferentially aligned fibers. Some numerical results are also presented. Under the case that is considered in this study, it is revealed that the CR disk offers much higher elastic fields than RR disk under all boundary conditions. For a composite rotating disk rotating at a constant speed, it will be better to align fibers along the radial directions, to use convergent disk profiles, and to locate the material having higher radial stiffness at the outer surface. It is also disclosed that the location of the maximum Von-Mises equivalent stress in fixed-free disks varies regarding the fiber orientation.

Determination of elastic resonance frequencies and eigenmodes using the method of fundamental solutions

In this paper, we present the method of fundamental solutions applied to the determination of elastic resonance frequencies and associated eigenmodes. The method uses the fundamental solution tensor of the Navier equations of elastodynamics in an isotropic material. The applicability of the the method is justified in terms of density results. The accuracy of the method is illustrated through 2D numerical examples for the disk and non trivial shapes.

On the Invariance of Governing Equations of Current Nonlocal Theories of Elasticity Under Coordinate Transformation and Displacement Gauge Change

The invariance of elastodynamic wave equations under coordinate transformations provides a way to achieve elastic wave cloaking. Under general coordinate transformations, it has been proved that the conventional elastodynamic wave equation (the Navier equation) changes its form. In addition to the conventional Navier equation, various nonlocal theories of elasticity have been developed to encompass more general descriptions of behaviour of solids. Whether the forms of the governing equations of these nonlocal theories of elasticity remain invariant under coordinate transformations has not been investigated. In this note, we examine the form-invariance of current nonlocal theories of elasticity, including the Mindlin 1964 theory, the strain gradient theory, the stress gradient theory, the peridynamic theory, the Kunin 1982 theory, the Eringen 1983 theory, and the Kroner 1967 theory under coordinate transformation and displacement gauge change. These theories are classified into three types and their invariance is examined in terms of three criteria. It is found that only the peridynamic theory, the Kunin 1982 theory, and the Eringen 1983 theory satisfy form-invariance. We further show that the operations of degeneration and a general coordinate transformation on nonlocal elasticity are not communicative. The results shed new light on the properties of the nonlocal theories of elasticity.

On the quasi-static effective behaviour of poroelastic media containing elastic inclusions

The aim of the present study is to derive the effective quasi-static behaviour of a composite medium, made of a poroelastic matrix containing elastic impervious inclusions. For this purpose, the asymptotic homogenisation method is used. On the local scale, the governing equations include Biot’s model of poroelasticity in the porous matrix and Navier equations in the inclusions, with elastic properties of the same order of magnitude. Biot’s diphasic model of poroelasticity is obtained on the macroscopic scale, but with effective parameters that are strongly impacted by the distribution of inclusions, even at low volume fraction. The impact on fluid flow is strictly geometrical, showing that the inclusions do not play the role of a porous network.

A Newtonian mechanics formulation for the vibration of translating and rotating elastic continua

In this paper, we present a Newtonian mechanics formulation for modeling the vibration of a convecting elastic continuum, i.e., a system characterized by mean kinematic translation or rotation with small superposed vibrations. The proposed Newtonian approach complements customary energy-based techniques and serves as a convenient means to validate and physically interpret their results. We develop the equations of motion and matching conditions in a continuum mechanics setting with respect to a stationary inertial reference frame. Interaction of the convecting continuum with discrete space-fixed elements (e.g., springs and dampers) is enabled, without introducing time-dependent coefficients, through the use of Eulerian kinematics and kinetics. Kinematic discontinuities inherent in these interactions are accommodated by employing a global (or integral) form of balance of linear momentum applied to a space-fixed control volume. A generalized form of Navier's equation of elastic wave propagation is derived, with unsteady, Coriolis, centripetal, and convective contributions to the inertia. The resulting formulation is applied to a broad class of translating and rotating systems – including spinning rings, axially moving strings and beams, and general three-dimensional elastic structures – and shown to successfully reconcile with existing energy-based derivations in the literature.

Analytical and numerical modeling of rock blasting operations using a two-dimensional elasto-dynamic Green's function

Rock media may be exposed to dynamic forces and time-dependent stresses. Dynamic phenomena in rock such as blasting and earthquakes need to propagate waves and dynamic stresses through the body. Therefore, stresses, strains, and displacements in rocks should be defined as time-dependent due to the dynamic phenomena. In this study, a general Green's function solution of an elastic wave propagation due to rock blasting has been analytically derived. The Navier's equations of motion have been assumed as governing equations and a general two-dimensional elastodynamic Green's function has been obtained in terms of displacement. The strain and stress fields related to the displacement Green's function have also been obtained using the theory of elasticity. Then, a generic two-dimensional problem containing an explosion in a blast hole has been solved using the resulting analytical solution. Moreover, a similar numerical modeling has been conducted using Finite Difference Method (FDM). Eventually, results from the analytical scheme have been compared with the finite difference numerical modeling and the valuable results from a previous experimental and numerical study.

An edge-based smoothed finite element method for wave scattering by an obstacle in elastic media

Scattering of a time-harmonic plane elastic wave by a rigid obstacle embedded in an isotropic homogeneous elastic medium has wide applications in science and engineering. This paper presents a novel method to study elastic wave scattering problems satisfying the Navier equation and Helmholtz equations with coupled boundary obtained by Helmholtz decomposition. We derive first smoothed Galerkin weak forms of Navier equation and Helmholtz equations to create effective smoothed finite element method (S-FEM) models. On the top of a three-noded triangular mesh, an edge-based S-FEM (ES-FEM-T3) combining with the perfectly matched layer (PML) technique is then established for solving the elastic wave scattering in bounded domains with PML to eliminate wave reflections. Some numerical experiments demonstrate that ES-FEM-T3 is more stable and accurate than the standard FEM for the elastic wave scattering.

General solutions for elasticity of transversely isotropic materials with thermal and other effects: A review

The well-known general solution for uncoupled thermoelasticity of isotropic bodies was proposed by Goodier, which has been utilized extensively since its birth in 1937. When the steady-state response is considered, the temperature field satisfies Laplace’s equation, and the corresponding elastic field can be expressed in terms of a harmonic function, giving rise to the Williams solution. However, for anisotropic bodies, there are no steady-state thermoelastic general solutions that are expressed in terms of (quasi)harmonic functions until 2000. This article presents a short review of the harmonic general solutions for uncoupled elasticity of transversely isotropic materials with thermal and other effects. These solutions are obtained by simply but forcedly combining the heat conduction equation with other governing equations (e.g., the Navier equations). We will show that the Williams solution as well as some other solutions all can be deduced as the special cases. Two application scenarios of the general solutions are also highlighted to demonstrate their elegance and versatility.

Stress analysis in rolling contact problem of a finite thickness FGM layer

The rolling contact problem of a non-homogeneous layer is considered here. The graded layer possesses a variable elastic modulus with an exponential distribution. The Poisons ratio is assumed to be constant. A rigid cylindrical indenter is rolling over the surface of the graded layer with a constant velocity. First, the Navier equations of equilibrium are solved in the Fourier domain. Later, the boundary and the continuity conditions are satisfied in order to extract the governing singular integral equations. The numerical solution of the integral equations is provided by means of the Gauss–Chebyshev integration method. Finally, the sensitivity of the solution is analyzed for the effective parameters namely: the stiffness ratio, the layer thickness and the coefficient of friction. The results indicate that a minimum value of the coating thickness is required to alleviate the severe stress gradients in the critical locations. If the coating thickness decreases by a 50% then the Von Mises stress will increases about 20%. Also, a softening graded layer can result in a lower stress level over the interface which may enhance the bonding strength.

Exact Thermomechanical Analysis of Functionally Graded (FG) Thick-Walled Spheres

Abstract The present study aims to provide a deeper understanding for the thermo-mechanical analysis of spheres made of non-homogeneous isotropic materials. To this end, Navier equations are solved analytically based on the spherically-symmetric plain-strain assumptions and closed-form formulas are proposed for the elastic fields in a simple-power-law graded spheres subjected to steady-state thermal and internal/external pressure loads. A comprehensive parametric study is then performed with both functionally-graded hypothetical and physical materials. Two benchmark examples are reconsidered with hypothetically chosen inhomogeneity indexes. Effects of inhomogeneity indexes are reviewed in these examples. Differently from the literature, thickness effects are also examined under separate and combined loads together with the thermo-mechanical behavioral differences in spheres and cylinders. Finally three physical metal-ceramic pairs are studied originally with appropriate inhomogeneity indexes which are defined as the inner surface is full ceramic and the outer surface is full metal. Results are presented in graphical and tabular forms.

Radiating and non-radiating sources in elasticity

In this work, we study the inverse source problem of a fixed frequency for the Navier equation. We investigate non-radiating external forces. If the support of such a force has a convex or non-convex corner or edge on its boundary, the force must be vanishing there. The vanishing property at corners and edges holds also for sufficiently smooth transmission eigenfunctions in elasticity. The idea originates from the enclosure method: an energy identity and a new type of exponential solution for the Navier equation.

The MAPS based on trigonometric basis functions for solving elliptic partial differential equations with variable coefficients and Cauchy–Navier equations

In this paper, we extended the method of approximate particular solutions (MAPS) using trigonometric basis functions to solve two-dimensional elliptic partial differential equations (PDEs) with variable-coefficients and the Cauchy–Navier equations. The new approach is based on the closed-form particular solutions for second-order differential operators with constant coefficients. For the Cauchy–Navier equations, a reformulation of the equations is required so that the particular solutions for the new differential operators are available. Five numerical examples are provided to demonstrate the effectiveness of the proposed method.

New strong formulation for material nonlinear problems based on the particle difference method

In this paper, we propose a strong form method for analyzing material nonlinear problems. The method utilizes the Particle Difference Method (PDM) which is classified as a strong form meshfree method and discretizes the governing equations based on a complete nodal computation without any integral formulation. The conventional strong form meshfree methods could not explicitly handle the nonlinear material model since they are mostly discretized based on the Navier's equation where kinematic variables are unified into displacement. To explicitly treat the nonlinear constitutive equation in the framework of strong formulation, a double derivative approximation is devised, which removes the need for the use of second order derivative approximation. The momentum equation is directly discretized through the double derivative approximation and is linearized by Newton's method to yield an iterative procedure for finding a converged solution. Stresses and internal variables are updated by the return mapping algorithm and efficiency of the iterative procedure is dramatically improved by the algorithmic tangent modulus. The consistency of the double derivative approximation was shown by the reproducing test. The accuracy and robustness of the developed nonlinear procedure were then verified through various inelastic material problems in one and two-dimensions.

Linear peridynamics for isotropic and anisotropic materials

A new approach is developed for linear peridynamics together with constitutive equations for isotropic and anisotropic materials. As a departure from the standard peridynamic theory, the linear constitutive equation in the form of micromodulus is determined by directly requiring the resulting peridynamic equation to converge to Navier's equation as the material horizon approaches 0 in the case of an isotropic material, and to a comparable elastodynamic equation in the case of an anisotropic material. As a result, new peridynamic governing equations complete with constitutive equations are obtained for isotropic as well as anisotropic elastic materials. As an application of the newly obtained peridynamic equations, a plane wave solution is analytically obtained and discussed, and dispersion curves are plotted for orthotropic peridynamic materials.

Flow distribution and mass transport analysis in cell geometries for redox flow batteries through computational fluid dynamics

In recent years, the development of new electrochemical cells for redox flow batteries (RFBs) has been of great interest. This paper presents the design and hydrodynamic evaluation of differing RFBs geometries, via computational tools, considering that experimental optimization is expensive and highly time consuming. A new geometry, similar to an electrochemical reactor that has a net-like spacer, is proposed as an alternative to conventional design. We applied a theoretical framework provided by the fluid dynamics equations of Navier–Stokes and Brinkman, considering a porous electrode in typical and modified cell geometries. The mass transport was evaluated via tracer molecule distribution at different time points after the tracer injection. The proposed geometry exhibited a more homogeneous flow distribution featuring an increased mean electrolyte velocity inside the electrode (six times higher than in a typical geometry featuring an interdigitated flow field). The increased velocity resulted in shorter retention times. Based on these results, the proposed geometry featuring a net-like spacer is a promising configuration for RFB applications. This work also provides a new approach for developing and characterizing RFBs.

Analysis of time‐domain elastic scattering by an unbounded structure

This paper is devoted to the analysis of the time‐domain elastic wave scattering problem in an unbounded structure. The transparent boundary condition is developed to reformulate the scattering problem into an initial‐boundary value problem in an infinite slab. The well posedness and stability are established for the reduced problem in both the frequency and time domains. Our proofs are based on the energy method, the Lax‐Milgram theorem, and the inversion theorem of the Laplace transform. Moreover, a priori estimates with explicit dependence on the time are achieved for the elastic displacement by taking special test functions for the time‐domain variational problems of the Navier equation.

Complex Fourier element shape functions for analysis of 2D static and transient dynamic problems using dual reciprocity boundary element method

In this paper, the boundary element method is reformulated using new complex Fourier shape functions for solving two-dimensional (2D) elastostatic and dynamic problems. For approximating the geometry of boundaries and the state variables (displacements and tractions) of Navier's differential equation, the dual reciprocity (DR) boundary element method (BEM) is reconsidered by employing complex Fourier shape functions. After enriching a class of radial basis functions (RBFs), called complex Fourier RBFs, the interpolation functions of a complex Fourier boundary element framework are derived. To do so, polynomial terms are added to the functional expansion that only employs complex Fourier RBF in the approximation. In addition to polynomial function fields, the participation of exponential and trigonometric ones has also increased robustness and efficiency in the interpolation. Another interesting feature is that no Runge phenomenon happens in equispaced complex Fourier macroelements, unlike equispaced classic Lagrange ones. In the end, several numerical examples are solved to illustrate the efficiency and accuracy of the suggested complex Fourier shape functions and in comparison with the classic Lagrange ones, the proposed shape functions result in much more accurate and stable outcomes.