Equations Of Linear Elasticity Research Articles

Scalability study of an implicit solver for coupled fluid-structure interaction problems on unstructured meshes in 3D

Fluid-structure interaction (FSI) problems are computationally very challenging. In this paper we consider the monolithic approach for solving the fully coupled FSI problem. Most existing techniques, such as multigrid methods, do not work well for the coupled system since the system consists of elliptic, parabolic and hyperbolic components all together. Other approaches based on direct solvers do not scale to large numbers of processors. In this paper, we introduce a multilevel unstructured mesh Schwarz preconditioned Newton–Krylov method for the implicitly discretized, fully coupled system of partial differential equations consisting of incompressible Navier–Stokes equations for the fluid flows and the linear elasticity equation for the structure. Several meshes are required to make the solution algorithm scalable. This includes a fine mesh to guarantee the solution accuracy, and a few isogeometric coarse meshes to speed up the convergence. Special attention is paid when constructing and partitioning the preconditioning meshes so that the communication cost is minimized when the number of processor cores is large. We show numerically that the proposed algorithm is highly scalable in terms of the number of iterations and the total compute time on a supercomputer with more than 10,000 processor cores for monolithically coupled three-dimensional FSI problems with hundreds of millions of unknowns.

Numerical Simulation of Transit-Time Ultrasonic Flowmeters by a Direct Approach.

This paper deals with the development of a computational code for the numerical simulation of wave propagation through domains with a complex geometry consisting in both solids and moving fluids. The emphasis is on the numerical simulation of ultrasonic flowmeters (UFMs) by modeling the wave propagation in solids with the equations of linear elasticity (ELE) and in fluids with the linearized Euler equations (LEEs). This approach requires high performance computing because of the high number of degrees of freedom and the long propagation distances. Therefore, the numerical method should be chosen with care. In order to minimize the numerical dissipation which may occur in this kind of configuration, the numerical method employed here is the nodal discontinuous Galerkin (DG) method. Also, this method is well suited for parallel computing. To speed up the code, almost all the computational stages have been implemented to run on graphical processing unit (GPU) by using the compute unified device architecture (CUDA) programming model from NVIDIA. This approach has been validated and then used for the two-dimensional simulation of gas UFMs. The large contrast of acoustic impedance characteristic to gas UFMs makes their simulation a real challenge.

Robust shape optimization of continuous structures via the level set method

This work proposes a stochastic shape optimization method for continuous structures using the level-set method. Such a method aims to minimize the expected compliance and its variance as measures of the structural robustness. The behavior of continuous structures is modeled by linear elasticity equations with uncertain loading and material. This uncertainty can be modeled using random variables with different probability distributions as well as random fields. The proper problem formulation is ensured by the proof of the existence colorrev of solution under certain geometrical constraints on the set of admissible shapes. The proposed method addresses the stochastic linear elasticity problem in its weak form obtaining the explicit expressions for the continuous shape derivatives. Some numerical examples are presented to show the effectiveness of the proposed approach.

Polynomial approach for modeling a piezoelectric disc resonator partially covered with electrodes

The frequency spectrum of a partially metallized piezoelectric disc resonator was studied using Legendre polynomials. The formulation, based on three-dimensional equations of linear elasticity, takes into account the high contrast between the electroded and non-electroded regions. The mechanical displacement components and the electrical potential were expanded in a double series of orthonormal functions and were introduced into the equations governing wave propagation in piezoelectric media. The boundary and continuity conditions were automatically incorporated into the equations of motion by assuming position-dependent physical material constants or delta-functions. The incorporation of electrical sources is illustrated. Structure symmetry was used to reduce the number of unknowns. The vibration characteristics of the piezoelectric discs were analyzed using a three-dimensional modelling approach with modal and harmonic analyses. The numerical results are presented as resonance and anti-resonance frequencies, electric input admittance, electromechanical coupling coefficient and field profiles of fully and partially metallized PIC151 and PZT5A resonator discs. In order to validate our model, the results obtained were compared with those published previously and those obtained using an analytical approach.

The simplest nonconforming mixed finite element method for linear elasticity in the symmetric formulation on [formula omitted]-rectangular grids

A family of mixed finite elements is proposed for solving the first order system of linear elasticity equations in any space dimension, where the stress field is approximated by symmetric finite element tensors. This family of elements has a perfect matching between the stress and the displacement. The discrete spaces for the normal stress τii, the shear stress τij and the displacement ui are span{1,xi}, span{1,xi,xj} and span{1}, respectively, on rectangular grids. In particular, the definition remains the same for all space dimensions. As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. In 1D, the element is nothing else but the 1D Raviart–Thomas element, which is the only conforming element in this family. In 2D and higher dimensions, they are new elements but of the minimal degrees of freedom. The total degrees of freedom per element are 2 plus 1 in 1D, 7 plus 2 in 2D, and 15 plus 3 in 3D. These elements are the simplest element for any space dimension.The well-posedness condition and the optimal a priori error estimate of the family of finite elements are proved. Numerical tests in 2D and 3D are presented to show a superiority of the new elements over others, as a superconvergence is exhibited and proved.

Wave Propagation in an Elastic Layer Interacting with a Viscous Liquid Layer

The Navier–Stokes three-dimensional linearized equations for a viscous fluid and the linear equations of classical elasticity for an elastic layer are used to plot dispersion curves and to study the propagation of acoustic waves over a wide frequency range. The effect of the viscosity of the fluid and the thickness of the elastic and liquid layers on the phase velocities and the damping factors of the modes is analyzed for both thin and thick solid layers. It is shown that for the thick elastic layer, there are certain thicknesses of the liquid layer and certain frequencies at which the effect of the viscosity of the fluid on the phase velocities and damping factors of all the modes is the weakest. It is also revealed that for a number of modes, there are certain frequencies and certain frequency ranges for which the effect of the viscosity of the fluid on the phase velocities and damping factors of the modes is strong. The approach developed and the results obtained allow identifying the limits of applicability of the model of ideal compressible fluid. The numerical results are presented in the form of plots and analyzed

Homogenization of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics

In this paper we present a derivation and multiscale analysis of a mathematical model for plant cell wall biomechanics that takes into account both the microscopic structure of a cell wall coming from the cellulose microfibrils and the chemical reactions between the cell wall’s constituents. Particular attention is paid to the role of pectin and the impact of calcium-pectin cross-linking chemistry on the mechanical properties of the cell wall. We prove the existence and uniqueness of the strongly coupled microscopic problem consisting of the equations of linear elasticity and a system of reaction-diffusion and ordinary differential equations. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive a macroscopic model for plant cell wall biomechanics.

Ion transport through deformable porous media: derivation of the macroscopic equations using upscaling

We study the homogenization (or upscaling) of the transport of a multicomponent electrolyte in a dilute Newtonian solvent through a deformable porous medium. The pore scale interaction between the flow and the structure deformation (modeled by linearized elasticity equations) is taken into account. After a careful adimensionalization process, we first consider so-called equilibrium solutions, in the absence of external forces, for which the velocity and diffusive fluxes vanish and the electrostatic potential is the solution of a Poisson–Boltzmann equation. When the motion is governed by a small static electric field and small hydrodynamic and elastic forces, we use O’Brien’s argument to deduce a linearized model. Then we perform the homogenization of these linearized equations for a suitable choice of time scale. It turns out that the deformation of the porous medium is weakly coupled to the electrokinetics system in the sense that it does not influence electrokinetics although the latter one yields an osmotic pressure term in the mechanical equations. As a consequence, the effective tensor satisfies Onsager properties, namely is symmetric positive definite.

Free vibration analysis of elastic elliptic cylinders with an eccentric elliptic cavity

An exact elastodynamic model based on Navier equations of linear elasticity is formulated to describe the three-dimensional natural oscillations of an elliptic cylinder of finite length with shear diaphragm end conditions, and containing an inner (coaxial) elliptical cavity of arbitrary size, location, and orientation. The formulation is based on Helmholtz decomposition theorem, the method of separation of variables in elliptical coordinates, and the translational addition theorems for Mathieu functions. The first three natural frequencies are calculated for selected cylinder lengths, cross-sectional aspect ratios, and cavity location/orientation parameters. Also, some representative 3D deformation mode shapes are depicted in vivid graphical form. The precision of solutions is checked through proper convergence studies, and the validity of results is verified with the aid of a commercial finite element package as well as by comparison with the available literature data. The presented exact Mathieu series solution is believed to be the first attempt on the vibrational characteristics of finite-length eccentric elliptical cylinders.

Shear-horizontal piezoelectric waves in an aluminum nitride film on a silicon substrate

ABSTRACTThe propagation of shear-horizontal waves in a piezoelectric film of aluminum nitride on a silicon substrate is studied. Three different electrode configurations are considered for thin film acoustic wave resonator application. A theoretical analysis is performed. The equations of linear piezoelectricity and anisotropic elasticity are used for the film and the substrate, respectively. Real and imaginary dispersion curves as well as electromechanical mode shapes are obtained. The effects of electrode configuration on the distribution of the electromechanical fields and the dispersion curves of long thickness-twist waves as well as energy trapping are examined.

A Highly Scalable Multilevel Schwarz Method with Boundary Geometry Preserving Coarse Spaces for 3D Elasticity Problems on Domains with Complex Geometry

We consider overlapping Schwarz algorithms for solving linear and nonlinear systems of equations arising from the finite element discretization of elasticity problems on unstructured meshes in three dimensions. The parallel scalability of Schwarz methods is determined almost completely by how the coarse space is constructed. In this paper, we introduce a low cost, boundary geometry preserving coarse mesh that shares the same boundary geometry with the fine mesh but has a better scalability than the coarse mesh obtained by uniformly coarsening the fine mesh. A new coarsening algorithm and a partitioning strategy are developed. We numerically show that a multilevel Schwarz method with the new coarse spaces is highly scalable, in terms of the total compute time, for solving some three-dimensional linear and nonlinear elasticity equations discretized on unstructured meshes with several hundreds of millions of unknowns and on a supercomputer with over 10,000 processor cores.

Mesoscale Models and Approximate Solutions for Solids Containing Clouds of Voids

For highly perforated domains the paper addresses a novel approach to study mixed boundary value problems for the equations of linear elasticity in the framework of mesoscale approximations. There are no assumptions of periodicity involved in the description of the geometry of the domain. The size of the perforations is small compared to the minimal separation between neighboring defects and here we discuss a class of problems in perforated domains, which are not covered by the homogenization approximations. The mesoscale approximations presented here are uniform. Explicit asymptotic formulas are supplied with the remainder estimates. Numerical illustrations, demonstrating the efficiency of the asymptotic approach developed here, are also given.

Modeling the elastic energy of alloys: Potential pitfalls of continuum treatments.

Some issues that arise when modeling elastic energy for binary alloys are discussed within the context of a Keating model and density-functional calculations. The Keating model is a simplified atomistic formulation based on modeling elastic interactions of a binary alloy with harmonic springs whose equilibrium length is species dependent. It is demonstrated that the continuum limit for the strain field are the usual equations of linear elasticity for alloys and that they correctly capture the coarse-grained behavior of the displacement field. In addition, it is established that Euler-Lagrange equation of the continuum limit of the elastic energy will yield the same strain field equation. This is the same energy functional that is often used to model elastic effects in binary alloys. However, a direct calculation of the elastic energy atomistic model reveals that the continuum expression for the elastic energy is both qualitatively and quantitatively incorrect. This is because it does not take atomistic scale compositional nonuniformity into account. Importantly, this result also shows that finely mixed alloys tend to have more elastic energy than segregated systems, which is the exact opposite of predictions made by some continuum theories. It is also shown that for strained thin films the traditionally used effective misfit for alloys systematically underestimate the strain energy. In some models, this drawback is handled by including an elastic contribution to the enthalpy of mixing, which is characterized in terms of the continuum concentration. The direct calculation of the atomistic model reveals that this approach suffers serious difficulties. It is demonstrated that elastic contribution to the enthalpy of mixing is nonisotropic and scale dependent. It is also shown that such effects are present in density-functional theory calculations for the Si-Ge system. This work demonstrates that it is critical to include the microscopic arrangements in any elastic model to achieve even qualitatively correct behavior.

Direct simulations of the linear elastic displacements field based on a lattice Boltzmann model

SummaryIn this paper, a lattice Boltzmann model for simulating linear elastic Lame equation is proposed. Differently from the classic lattice Boltzmann models, this lattice Boltzmann model is based on displacement distribution function in lattice Boltzmann equation. By using the technique of the higher‐order moments of equilibrium distribution functions and a series of partial differential equations in different time scales, we obtain the Lame equation with fourth‐order truncation errors. Based on this model, some problems with small deflection are simulated. The comparisons between the numerical results and the analytical solutions are given in detail. The numerical examples show that the lattice Boltzmann model can be used to solve problems of the linear elastic displacement field with small deflection. Copyright © 2015 John Wiley & Sons, Ltd.

Realizing the Willis equations with pre-stresses

This paper proves that the linear elastic behavior of the material with inhomogeneous pre-stresses can be described by the Willis equations. In this case, the additional terms in the Willis equations, compared with the classical linear elastic equations for homogeneous media, are related to the gradient of pre-stresses. In this way, the material length scale is naturally incorporated in the framework of continuum mechanics. All these findings also coincide with the results of transformation elastodynamics, so that they can meet the requirement of the principle of material objectivity and the principle of general invariance.

Hardy space infinite elements for time-harmonic two-dimensional elastic waveguide problems

We consider time-harmonic linear elasticity equations in domains containing two-dimensional semi-infinite strips. Since for such problems there exist modes with different signs of group and phase velocity, standard perfectly matched layer (PML) as well as standard Hardy space infinite element methods fail.We apply a recently developed infinite element method for a physically correct discretization of such waveguide problems which is based on a Laplace transform in propagation direction. In the Laplace domain the space of transformed solutions can be separated into a sum of a space of incoming and a space of outgoing functions where both function spaces are certain Hardy spaces. The Hardy space is chosen such that the construction of a simple infinite element is possible.The method does not use a modal separation and works on domains of frequencies. On those domains the involved operators are frequency independent and hence lead to linear eigenvalue problems when computing resonances. Numerical experiments containing convergence tests and resonance problems are included.

On the compatibility equations of nonlinear and linear elasticity in the presence of boundary conditions

We use Hodge-type orthogonal decompositions for studying the compatibility equations of the displacement gradient and the linear strain with prescribed boundary displacements. We show that the displacement gradient is compatible if and only if for any equilibrated virtual first Piola–Kirchhoff stress tensor field, the virtual work done by the displacement gradient is equal to the virtual work done by the prescribed boundary displacements. This condition is very similar to the classical compatibility equations for the linear strain. Since these compatibility equations for linear and nonlinear strains involve infinite-dimensional spaces and consequently are not easy to use in practice, we derive alternative compatibility equations, which are written in terms of some finite-dimensional spaces and are more useful in practice. Using these new compatibility equations, we present some non-trivial examples that show that compatible strains may become incompatible in the presence of prescribed boundary displacements.

Resonances of a submerged fluid-filled spherically isotropic microsphere with partial-slip interface condition

Motivated by the numerous applications of spherical shell models in micro and nano scales (such as microbubbles, bacterial cells, and viral capsids), we have considered the axisymmetric free vibrations of a spherically isotropic fluid-filled thick microspherical shell suspended in another unbounded fluid. A partial-slip condition is considered at the solid-fluid interface(s). Three-dimensional linear elasticity equations for the spherically isotropic shell dynamics and linearized Navier-Stokes equations for the two compressible viscous fluids are used in the analysis. The eigenvalue problem is discretized and solved to find the resonances and quality factors. A perfectly matched layer technique is used to separate the solid driven spectrum from the boundary reflecting spectrum. An example of air filled polymer shell suspended in water is presented. The added mass effect and partial-slip condition from water (air) on the frequencies and quality factors are found to be significant (negligible). Spherical isotropy is found to have major influence on the low frequency and large meridional wave number region of the resonance spectrum. High quality eigenmodes are observed due to very small viscous penetration depth compared to the shell size. In the thin-shell limit, the eigenvalue problem can have only two modes of vibration for any meridional wave number greater than or equal to two. This explains the reason for the second resonance frequency found for the quadrupole shape oscillations of various bacterium cells in the earlier work. The partial-slip condition is found to have very small influence on the first few modes of vibration. Surface tension is found to have significant influence only on the lowest frequency trend of the eigenspectrum. Perfectly matched layer technique used in the present analysis is found to be very effective in handling the boundary truncated problems.

Nonlinear deformations of piezoelectric composite beams

This study presents large deformations of slender elastic and viscoelastic beams with multiple piezoelectric patches attached on their top and bottom surfaces. The slender beams can undergo large in-plane 2-D deformations due to electric fields applied through the piezoelectric patches and mechanical actuations. A nonlinear electro-mechanical constitutive equation is considered for the piezoelectric patches, while linear elastic and viscoelastic constitutive equations are used for the beams. Reissner’s finite-deformation beam theory is adopted in formulating the large 2-D deformation, and modified in order to incorporate the deformation due to the electric field input. For an elastic beam, closed form solutions are obtained for the deformations of the beam under electric field actuation, while a nonlinear shooting method is used to analyze the deformation of the beam under both electrical- and mechanical stimuli. For viscoelastic beams, time-dependent deformations of the beams under electric field actuations through the piezoelectric patches are solved numerically. By applying electric fields with different amplitude at different locations of patches, desired deformed shapes in active flexible beams can be achieved, which is useful for analyses and designs of active foldable systems. This study also highlights the effect of viscoelastic materials on the shape changes in foldable electro-active beams.

Morphological Evolution in Heteroepitaxial Thin Film Structures at the Nanoscale

The aim of this study is to resolve the phenomenon of formation of mesoscopic structures on the surface of heteroepitaxial thin film system due to surface diffusion by considering the effects of both surface and interface stresses. Elastic stress field caused by curved surface is solved by using the constitutive equations of linear elasticity for the bulk and surface phases. Based on the method of superposition, a boundary perturbation technique, Goursat-Kolosov complex potentials and Muskhelishvili representations, the boundary value problem is reduced to the successive solution of a system of singular and hypersingular integral equations for any order of approximation. This solution and thermodynamic approach allows us to derive a governing equation which gives the amplitude changing of a surface roughness with time.