Navier Equations Research Articles

Parallelizing viscoelastic shear wave simulations with MPI and CUDA

Three dimensional (3D) simulations of shear wave elasticity imaging (SWEI) are performed by computing a 3D push beam in FOCUS, the “Fast Object-oriented C++ Ultrasound Simulator” (https://www.egr.msu.edu/~fultras-web) followed by the superposition of Green's function solutions to Navier's equation in viscoelastic media. Results show that these Green's function calculations are particularly amenable to parallel implementations that combine MPI and CUDA.

Elastic wave near-cloaking

Cloaking elastic waves has, in contrast to the cloaking of electromagnetic waves, remained a fundamental challenge: the latter successfully uses the invariance of Maxwell’s equations, from which the field of transformational optics has emerged, whereas the elastic Navier equations are not invariant under coordinate transformations. Our aim is to overcome this challenge, at least in practical terms, and thereby unlock applications in mechanics, ultrasound, vibration mitigation, non-destructive evaluation and elastic wave control. We achieve near-cloaking by recognizing that, despite the lack of invariance, a decoupling into a system of form invariant potential equations together with a quantifiable approximation, can be used effectively in many cases to control the flow of elastodynamic waves. Here, in particular we focus on the efficiency and practicability of the proposed near-cloaking which is illustrated using carpet cloaks to hide surface defects from incoming compressional and shear in-plane waves and from surface elastic Rayleigh waves.

Analyzing the influence of a particle's linear and angular velocity on the equations of liquid motion

This paper has analyzed the equation of motion in terms of stresses (Navier), as well as its two special cases for an incompressible viscous current. One is the Stokes (Navier-Stokes) equation, and the other was derived with fewer restrictions. It has been shown that the Laplace equation of linear velocity can be represented as a function of two variables ‒ the linear and angular speed of particle rotation. To describe the particle acceleration, all motion equations employed a complete derivative from speed in the Gromeka-Lamb form, which depends on the same variables. Taking into consideration the joint influence of linear and angular velocity allows solving a task of the analytical description of a turbulent current within the average model. A given method of analysis applies the provision of general physics that examines the translational and rotational motion. The third type of mechanical movement, oscillatory (pulsation), was not considered in the current work. A property related to the Stokes equation decomposition has been found; a block diagram composed of equations and conditions has been built. It is shown that all equations for viscous liquid have their own analog in a simpler model of non-viscous fluid. That makes it easier to find solutions to the equations for the viscous flow. The Stokes and Navier equations were used to solve two one-dimensional problems, which found the distribution of speed along the normal to the surface at the flow on a horizontal plate and in a circular pipe. Both solution methods produce the same result. No solution for the distribution of speed along the normal to the surface in a laminar sublayer could be found. A relevant task related to the mathematical part is to solve the problem of closing the equations considered. A comparison of the theoretical and empirical equations has been performed, which has made it possible to justify the assumption that a rarefied gas is the Stokes liquid

The treatment of constraints due to standard boundary conditions in the context of the mixed Web-spline finite element method

PurposeThis paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous and non-homogeneous; Dirichlet boundary conditions. This paper considered the two-dimensional linear elasticity equation of Navier–Lamé with the condition CAB. The latter allows to have a total insertion of the essential boundary condition in the linear system obtained; without using a numerical method as Lagrange multiplier. This study have developed mixed finite element; method using the B-splines Web-spline space. These provide an exact implementation of the homogeneous; Dirichlet boundary conditions, which removes the constraints caused by the standard; conditions. This paper showed the existence and the uniqueness of the weak solution, as well as the convergence of the numerical solution for the quadratic case are proved. The weighted extended B-spline; approach have become a much more workmanlike solution.Design/methodology/approachIn this paper, this study used the implementation of weighted finite element methods to solve the Navier–Lamé system with a new boundary condition CA, B (Koubaiti et al., 2020), that generalises the well-known basis, especially the Dirichlet and the Neumann conditions. The novel proposed boundary condition permits to use a single Matlab code, which summarises all kind of boundary conditions encountered in the system. By using this model is possible to save time and programming recourses while reap several programs in a single directory.FindingsThe results have shown that the Web-spline-based quadratic-linear finite elements satisfy the inf–sup condition, which is necessary for existence and uniqueness of the solution. It was demonstrated by the existence of the discrete solution. A full convergence was established using the numerical solution for the quadratic case. Due to limited regularity of the Navier–Lamé problem, it will not change by increasing the degree of the Web-spline. The computed relative errors and their rates indicate that they are of order 1/H. Thus, it was provided their theoretical validity for the numerical solution stability. The advantage of this problem that uses the CA, B boundary condition is associated to reduce Matlab programming complexity.Originality/valueThe mixed finite element method is a robust technique to solve difficult challenges from engineering and physical sciences using the partial differential equations. Some of the important applications include structural mechanics, fluid flow, thermodynamics and electromagnetic fields (Zienkiewicz and Taylor, 2000) that are mainly based on the approximation of Lagrange. However, this type of approximation has experienced a great restriction in the level of domain modelling, especially in the case of complicated boundaries such as that in the form of curvilinear graphs. Recently, the research community tried to develop a new way of approximation based on the so-called B-spline that seems to have superior results in solving the engineering problems.

Spherical oscillations of encapsulated microbubbles: Effect of shell compressibility and anisotropy.

We introduce a model that describes spherical oscillations of encapsulated microbubbles in an unbounded surrounding fluid. A Rayleigh-Plesset-like equation is derived by coupling the Navier-Stokes equation that describes fluid dynamics with the Navier equation that describes solid dynamics via the internal/external boundary conditions. While previous models were restricted to incompressible isotropic shells, the solid shell is modeled here as a compressible viscoelastic isotropic material and then generalized to an anisotropic material. The exact value of the resonance frequency is calculated analytically, and the damping constant is computed in the approximation of weak damping. A correction of the widely used Church model for incompressible shells is evidenced, and the effects of shell compressibility and anisotropy are discussed.

Analytical evaluation of transformation matrix between functional spaces and its application in obtaining exact solutions of thermo-mechanical stresses in disks

Thermo-mechanical stresses developed in a disk, inner boundary of which is subject to the nonstationary thermo-mechanical loadings due to variation in temperature and pressure inside the disk is considered in the presented paper. Problems of theoretical analysis are divided and described as follows: The first problem implies analytical solution of the heat conduction equation with nonhomogeneous boundary conditions of Robin type, and the second problem deals with obtaining analytical solution of dynamic equation of displacement for nonisothermal elastic body (Navier equation) with traction boundary conditions in the plane strain case of uncoupled thermo-elasticity. Finite Hankel transform is used to solve the problems. Two sets of orthogonal basis functions necessary for integral transform are obtained. The first one corresponds to the boundary conditions written for the heat conduction equation, and the second fits the boundary conditions for Navier equation. Integrals defining elements of transformation matrix between the two sets of basis functions are evaluated analytically. In general, convenient method of obtaining analytical solution of linear hyperbolic equation with nonhomogeneous term, which itself is solution of nonhomogeneous parabolic equation, is elaborated within the presented work. Using finite difference method, discrete counterparts of the described problems are constructed and, via the built-in ordinary differential equation (ODE) solvers in MATLAB, numerical solutions are obtained, which exhibit excellent accordance with analytical ones.

TO QUESTION OF APPLICATION OF HYDRODYNAMIC ANALOGY FOR PROCEDURE OF CALCULATION OF PARAMETERS OF TRANSPORT STREAMS

A mathematical modeling of traffic flow for today is rather intricate and actual problem. The most perfect mathematical models of traffic flow are described by equations of mathematical physics. In the theory of traffic flow there are various approaches to classification of their mathematical models. One of widespread is classification on macroscopic models and microscopic models. The whole group of transport vehicles, that is described by the corresponding parameters of motion, is examined in macroscopic models. Microscopic models are based on conception of safe distance to the leader. The most known models are: model of optimal speed, following by a leader model, Treibe’s model of clever driver. Historically one of the first macroscopic models is a model of Lighthill – Whitham – Richards (LWR). In it the stream of motor-car transport vehicles is considered as an unidimensional stream of compressible liquid. In a model LWR is accepted, that an unambiguous interconnection exists between speed and fluid flow density, and the laws of maintenance of mass are satisfied. Other macroscopic models based on analogues of a traffic flow to the hydrodynamic features of compressible liquid flow exist. It is a model of Tanaka, Whitham. Payne et al. The model of Helbing Eeler Navier Stokes is proposed in 1995. In this model to the system of Payne’s equations the third equation that represents the law of conservation of energy for variation of speed is added. In the second equation (law of momentum conservation) an additional component is considered that allows to take into account variation of speed. It should be noted that for the system of Navier – Stokes equations it is not known how to formulate the initial Cauchy boundary value problem in order the global solution is unique for all times. The main difference between the hydrodynamic models of traffic flows from the corresponding hydrodynamic analogs is the formation of the right-hand side of the equations. This refers to the correct notation, as a rule, of hyperbolic systems of equations and their diffusion analogs. The difficulties arising in the description of the traffic flow are similar to those that arise in the description of the turbulent motion of a liquid. The aim of work is a construction of mathematical model, numerical method, algorithm for obtaining numerical solutions and creation of software for studying the dynamics of traffic flows. The task of modeling of traffic flows of motor-car transport vehicles is examined in the article. For description of physical process the system of equations of Navier – Stokes is used. Methodology, numerical algorithm, and software, is worked out. For numerical integration of the systems of differential equations, finite volume method is used. The developed methodology was tested. Based on the results of numerical calculations, a fundamental traffic flow diagram has been constructed.

Analysis of Concrete Tensile Failure using Dynamic Particle Difference Method under High Loading Rates

This study presents the use of the dynamic particle difference method (PDM) to analyze tensile failure in concrete subjected to high loading rates. In general, strong form-based meshfree methods suffer from limitations pertaining to the material modeling of concrete because concrete exhibits both softening and damage behaviors that initiate crack growth under an impact load. These methods are generally based on the direct discretization of the governing equations, such as Navier's equation, which involves second-order differentiation. However, conventional material models are based on the first-order derivatives of displacement. The newly developed dynamic PDM can effectively address the limitations of material modeling using a combination of first-order derivative approximations. This circumvents the requirement for high-order derivative approximations, which are essential in strong formulations, such as the finite difference method and point collocation method. The strain rate effect caused by an extremely high loading speed was successfully modeled by accurately reflecting the energy dissipations that result from the cohesive property of the concrete and brittle cracking. Although the developed method incorporates the elastic constitutive relation, it enables the effective modeling of these nonlinear effects. In addition, it can reduce the computational effort. The proportional damping algorithm simulates the effect of velocity in the equation of motion and the cohesion effect in the concrete material. The damage model and the visibility criterion adequately handle crack initiation and propagation in the concrete member. Furthermore, it is noteworthy that the final discrete forms of the dynamic PDM are similar to the integrands of the weak form in the conventional finite element formulation. We ascertained that the stiffness and mass proportional damping effects are related to the inertia and strain of the material, respectively. It was confirmed that the location and direction of crack propagation in concrete varied with the strain rate. Hence, the accuracy and robustness of the proposed method were successfully verified by simulation, and the strain-rate dependency of concrete fracture was efficiently simulated using the proposed method.

Effect of predominantly unidirectional flow of a viscous liquid between solid walls

The problem is considered on a flow of a viscous liquid in the presence of solid bodies – two walls and a free plate the boundary of which is permeable for a liquid – under oscillatory influences. The formulation of the problem includes the equation of the plate motion, the equations of Navier–Stokes and a continuity and conditions at solid boundaries of the liquid. The new hydro-mechanical effect is revealed which consists in that in the absence of a predominant direction in space free parts of the hydro-mechanical system at a background of oscillations perform a unidirectional, steady motion.

Influence of Thermoelastic Phenomena on the Energy Conservation in Non-Contacting Face Seals

The purpose of this study was to develop a mathematical model for non-contacting face seals to analyze how their performance is affected by thermoelastic phenomena. The model was used to solve thermal conductivity and thermoelasticity problems. The primary goal was to calculate the values of thermal deformations of the sealing rings in a non-contacting face seal with a flexibly mounted rotor (FMR) for a turbomachine. The model assumes the conversion of mechanical energy into heat in the fluid film. The heat flux generated in the fluid film is transferred first to the sealing rings and then to the fluid surrounding them. Asymmetric distribution of temperature within the sealing rings leads to the occurrence of thermal stresses and, consequently, a change in the geometry of the rings. The model is solved analytically. The distributions of temperature fields for the sealing rings in the cross-sections are calculated using the Fourier-Bessel series as a superficial function of two variables (r,z). The thermoelasticity problems described by the Navier equations are solved by applying the Boussinesq harmonic functions and Goodier’s thermoelastic displacement potential function. The proposed method involves solving various theoretical and practical problems of thermoelasticity in FMR-type non-contacting face seals. The solution of the mathematical model was made use of analytical methods, and the most important obtained results are presented in graphical form, such as the temperature distributions and axial thermal distortions in cross-sections of the rings. The calculated thermal deformations of the sealing rings are used to determine the most important seal performance parameters such as the leakage rate and power loss. The article also presents a multi-criteria analysis of seal rings materials and geometry, which makes it easier to choose the type of materials used for the sliding rings.

Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation

In this work, we study reproducing kernel (RK) collocation method for peridynamic Navier equation. In the first part, we apply a linear RK approximation to both displacement and dilatation, and then back-substitute dilatation and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit for a fixed nonlocal interaction length and also to the local limit as nonlocal interactions vanish. The stability is shown by comparing the collocation scheme with the standard Galerkin scheme using Fourier analysis. In the second part, we apply the RK collocation to the quasi-discrete peridynamic Navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed. The analysis is carried out on a special family of rectilinear Cartesian grids for the RK collocation method with a designated kernel with finite support. We assume the Lamé parameters satisfy λ≥μ to avoid extra assumptions on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.

Inverse random source problems for time-harmonic acoustic and elastic waves

This paper concerns the random source problems for the time-harmonic acoustic and elastic wave equations in two and three dimensions. The goal is to determine the compactly supported external force from the radiated wave field measured in a domain away from the source region. The source is assumed to be a microlocally isotropic generalized Gaussian random function such that its covariance operator is a classical pseudo-differential operator. Given such a distributional source, the direct problem is shown to have a unique solution by using an integral equation approach and the Sobolev embedding theorem. For the inverse problem, we demonstrate that the amplitude of the scattering field averaged over the frequency band, obtained from a single realization of the random source, determines uniquely the principle symbol of the covariance operator. The analysis employs asymptotic expansions of the Green functions and microlocal analysis of the Fourier integral operators associated with the Helmholtz and Navier equations.

Calculation of the BEM Integrals on a Variable Grid With the FFT

In this study, an exemplary application of the pFFT is shown for the 2D Navier equation for a linear elastic continuum. Using this example, it is illustrated how the pFFT might be extended in order to decrease the computational complexity of the method. In the standard pFFT approach, all panel influences which are not calculated directly are obtained using a single regular grid. In the present study, a variable gird is suggested to obtain these influences. It is outlined how it is possible to apply the FFT on each level of this variable grid by rearranging segments of the shape boundary. A brief example is presented which indicates feasibility of the concept.

Two-dimensional elastodynamic and free vibration analysis by the method of fundamental solutions

In this paper we will present a formulation for solving elastodynamic problems in 2D using the method of fundamental solutions (MFS). The governing equation for the displacement in the elastodynamic problem is expressed by the Navier equation with an additional non-homogeneous inertial term, which involves the second derivative of displacement with respect to time. The inertial term is approximated by the Houbolt finite difference formula. In each time step, the solution for the displacement is separated into a homogeneous solution and a particular solution. The homogeneous solution is constructed in terms of the fundamental solutions of the elastostatic problem. The particular solution corresponds to the non-homogeneous inertial term, which is obtained by approximating the inertial term with radial basis functions (RBFs). In addition, a formulation based on the MFS for free vibration analysis of 2D bodies is presented in this paper. By solving several numerical examples, we have demonstrated that the proposed methods are robust and accurate. The convergence of the results with respect to the number of collocation points, the number of internal points, the value of the RBF parameter, and the time step size is also studied.

Elastic scattering from rough surfaces in three dimensions

Consider the elastic scattering of a plane or point incident wave by an unbounded and rigid rough surface. The angular spectrum representation (ASR) for the time-harmonic Navier equation is derived in three dimensions. The ASR is utilized as a radiation condition to the elastic rough surface scattering problem. The uniqueness is proved through a Rellich-type identity for surfaces given by uniformly Lipschitz functions. In the case of flat surfaces with local perturbations, an equivalent variational formulation is deduced in a truncated bounded domain and the existence of solutions are shown for general incoming waves. The main ingredient of the proof is the radiating behavior of the Green tensor to the first boundary value problem of the Navier equation in a half-space.

Torsional vibration of the porous nanotube with an arbitrary cross-section based on couple stress theory under magnetic field

This article presented a solution for torsion vibration a nanotube made of pores material. Based on the porosity distribution, the material properties are assumed to vary according to a function along radius of nanotube. Moreover, the cross-section area of nanotube varied in the longitudinal direction by nonlinear function. As for the torque effect of the axial magnetic field, the well-known Maxwell’s relation is used. Couple stress theory is employed to study the influence of small-scale on torsional vibration of nanotube. The Navier equation and boundary conditions of the size-dependent porous nanotube were derived by the Hamilton principle. These equations were solved by employing the generalized differential quadrature method. Comparison between the results of the present work with the other paper reveals the accuracy of this study. To the best of authors’ knowledge, so far all previous torsional vibration of nanotube address the case of ignoring porosity. The novelty of this work is to present a solution by taking into account the existence of porosity. Finally, numerical results are presented to study the small scale effect and porosity on the frequency of the porous nanotube.

Static Torsion of Bi-Directional Functionally Graded Microtube Based on the Couple Stress Theory Under Magnetic Field

This paper presents is a solution for static torsion in a microtube made of bi-directional functionally graded materials (BDFGMs). The material properties are assumed to vary according to the arbitrary function along radius and length of microtube. As for the torque effect of the axial magnetic field, the well-known Maxwell’s relation is used. Couple stress theory is employed to study the influence of small-scale on static torsion of microtube. The Navier equation and boundary conditions of the size-dependent BDFG microtube were derived by the minimum potential energy. These equations were solved by employing the generalized differential quadrature method (GDQM). Comparison between the results of this work with the analytical method for static torsion of microtube made of one direction FG material reveals the accuracy of this study. Finally, numerical results are presented to study the small-scale effect and heterogeneity constants on the static torsion of the bi-directional functionally graded microtube.

Viscosity Measurement by the “Oscillating Drop Method”: The Case of Strongly Damped Oscillations

By the measurement of the frequency and damping time of surface oscillations, excited by a short pulse on a freely floating liquid droplet, the surface tension and viscosity of the liquid can under certain conditions contactlessly be determined. The conventional physical models linking these material properties to the corresponding measurement quantities are the Rayleigh and the Lamb formula. However, both these equations are derived under the assumption of undamped or very weakly damped oscillations. In the following the basic characteristic equation of the linear Navier–Stokes equation, representing a corresponding physical model for the whole damping range, is theoretically investigated. A comparison of its accurate results with those of Rayleigh and Lamb reveal, that the Rayleigh formula provides, even in the case of strongly damped oscillations, an approximation which is still good enough for most practical purposes, whereas the Lamb formula deviates by more than 10 % from the accurate value. An improvement of the Lamb formula is derived from the characteristic equation which extends its practical applicability also to strongly damped oscillations.

Natural Domain Decomposition Algorithms for the Solution of Time-Harmonic Elastic Waves

We study for the first time Schwarz domain decomposition methods for the solution of the Navier equations modeling the propagation of elastic waves. These equations in the time-harmonic regime are difficult to solve by iterative methods, even more so than the Helmholtz equation. We first prove that the classical Schwarz method is not convergent when applied to the Navier equations and can thus not be used as an iterative solver, only as a preconditioner for a Krylov method. We then introduce more natural transmission conditions between the subdomains and show that if the overlap is not too small, this new Schwarz method is convergent. We illustrate our results with numerical experiments, both for situations covered by our technical two subdomain analysis and situations that go far beyond, including many subdomains, cross points, heterogeneous materials in a transmission problem, and Krylov acceleration. Our numerical results show that the Schwarz method with adapted transmission conditions leads systematically to a better solver for the Navier equations than the classical Schwarz method.

Analysis of Convective Transport of Temperature-Dependent Viscosity for Non-Newtonian Erying Powell Fluid: A Numerical Approach

Non-Newtonian is a type of fluid that does not comply with the viscosity under the Law of Newton and is being widely used in industrial applications. These include those related to chemical industries, cosmetics manufacturing, pharmaceutical field, food processing, as well as oil and gas activities. The inability of the conventional equations of Navier–Stokes to accurately depict rheological behavior for certain fluids led to an emergence study for non-Newtonian fluids’ models. In line with this, a mathematical model of forced convective flow on non-Newtonian Eyring Powell fluid under temperature-dependent viscosity (TDV) circumstance is formulated. The fluid model is embedded with the Newtonian heating (NH) boundary condition as a heating circumstance and is assumed to move over a stretching sheet acting vertically. Using appropriate similarity variables, the respective model was converted into ordinary differential equations (ODE), which was later solved utilizing the Keller box approach. The present model is validated by comparing the existing output in literature at certain special limiting cases, where the validation results display a firm agreement. The current outputs for the proposed model are shown in tabular and graphical form for variation of skin friction plus Nusselt number, velocity and temperature distribution, respectively.