Equations Of Linear Elasticity Research Articles

Programming finite element methods with weighted B-splines

We discuss the main components of the recently developed FEMB program package, which implements finite element methods with weighted B-splines for basic linear elliptic boundary value problems in two and three dimensions. A three-dimensional implementation without topological restrictions has not been available before. We describe in particular the mathematical background for the recursive quadrature/cubature over boundary cells and explain how to utilize the regular data structure of uniform B-splines efficiently. Considering the Lamé–Navier equations of linear elasticity as a typical example, we illustrate the performance of the main FEMB routines. The numerical tests confirm that, due to the new integration routines, the weighted B-spline solvers have become considerably more efficient.

Homogenization of a multiscale viscoelastic model with nonlocal damping, application to the human lungs

We are interested in the mathematical modeling of the deformation of the human lung tissue, called the lung parenchyma, during the respiration process. The parenchyma is a foam-like elastic material containing millions of air-filled alveoli connected by a tree-shaped network of airways. In this study, the parenchyma is governed by the linearized elasticity equations and the air movement in the tree by the Poiseuille law in each airway. The geometric arrangement of the alveoli is assumed to be periodic with a small period ε > 0. We use the two-scale convergence theory to study the asymptotic behavior as ε goes to zero. The effect of the network of airways is described by a nonlocal operator and we propose a simple geometrical setting for which we show that this operator converges as ε goes to zero. We identify in the limit the equations modeling the homogenized behavior under an abstract convergence condition on this nonlocal operator. We derive some mechanical properties of the limit material by studying the homogenized equations: the limit model is nonlocal both in space and time if the parenchyma material is considered compressible, but only in space if it is incompressible. Finally, we propose a numerical method to solve the homogenized equations and we study numerically a few properties of the homogenized parenchyma model.

Solvability analysis and numerical approximation of linearized cardiac electromechanics

This paper is concerned with the mathematical analysis of a coupled elliptic–parabolic system modeling the interaction between the propagation of electric potential and subsequent deformation of the cardiac tissue. The problem consists in a reaction–diffusion system governing the dynamics of ionic quantities, intra- and extra-cellular potentials, and the linearized elasticity equations are adopted to describe the motion of an incompressible material. The coupling between muscle contraction, biochemical reactions and electric activity is introduced with a so-called active strain decomposition framework, where the material gradient of deformation is split into an active (electrophysiology-dependent) part and an elastic (passive) one. Under the assumption of linearized elastic behavior and a truncation of the updated nonlinear diffusivities, we prove existence of weak solutions to the underlying coupled reaction–diffusion system and uniqueness of regular solutions. The proof of existence is based on a combination of parabolic regularization, the Faedo–Galerkin method, and the monotonicity-compactness method of Lions. A finite element formulation is also introduced, for which we establish existence of discrete solutions and show convergence to a weak solution of the original problem. We close with a numerical example illustrating the convergence of the method and some features of the model.

A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations

We consider multigrid methods for discontinuous Galerkin $$H({\text {div}},\Omega )$$H(div,Ω)-conforming discretizations of the Stokes and linear elasticity equations. We analyze variable V-cycle and W-cycle multigrid methods with nonnested bilinear forms. We prove that these algorithms are optimal and robust, i.e., their convergence rates are independent of the mesh size and also of the material parameters such as the Poisson ratio. Numerical experiments are conducted that further confirm the theoretical results.

Interface deformations due to counter-rotating vortices: Viscous versus elastic media.

Capillary forces determine the shape of a liquid interface. Although often not considered, elastic solids with a free surface are also subjected to surface forces and these become important for materials of low Young's modulus. Here we consider two equivalent problems where a capillary free surface deforms due to vortices: (i) in a steady viscous flow [solved by Jeong and Moffatt, J. Fluid Mech. 241, 1 (1992)], and (ii) in an elastic medium. The equations of linear incompressible elasticity and viscous flow are strictly identical, and the two-dimensional problems that we consider are solved using complex variable methods. Despite the similarity, the kinematics of the free surface is very different for the viscous and elastic cases. We show for the present problem that these kinematics result in displacement and velocity fields of different topology. Unexpectedly, the resulting surface deflections are even of opposite sign.

Unified Analysis with Mixed Finite Element Formulation for Acoustic-Porous-Structure Multiphysics System

This research aims to develop a novel unified analysis method for an acoustic-porous-structure multiphysics interaction system when the porous medium is modeled by the empirical Delany–Bazley formulation. Multiphysics analysis of acoustic structure interaction is commonly performed by solving the linear elasticity and Helmholtz equations separately and enforcing a mutual coupling boundary condition. If the pressure attenuation from a porous material is additionally considered, the multiphysics analysis becomes highly intricate, because three different media (acoustic, porous, and elastic structures) with different governing equations and interaction boundary conditions should be properly formulated. To overcome this difficulty, this paper proposes the application of a novel mixed formulation to consider the mutual coupling effects among the acoustic, fibrous (porous), and elastic structure media. By combining the mixed finite element formulation with the Delany–Bazley formulation, a multiphysics simulation of sound propagation considering the coupling effects among the three media can be easily conducted. To show the validity of the present unified approach, several benchmark problems are considered.

Homogenization of a Model for the Propagation of Sound in the Lungs

In this paper, we are interested in the mathematical modeling of the propagation of sound waves in the lung parenchyma, which is a foam-like elastic material containing millions of air-filled alveoli. In this study, the parenchyma is governed by the linearized elasticity equations, and the air by the acoustic wave equations. The geometric arrangement of the alveoli is assumed to be periodic with a small period $\varepsilon>0$. We consider the time-harmonic regime forced by vibrations induced by volumic forces. We use the two-scale convergence theory to study the asymptotic behavior as $\varepsilon$ goes to zero and prove the convergence of the solutions of the coupled fluid-structure problem to the solution of a linear-elasticity boundary value problem.

Dynamics with Matrices Possessing Kronecker Product Structure

In this paper we present an application of Alternating Direction Implicit (ADI) algorithm for solution of non-stationary PDE-s using isogeometric finite element method. We show that ADI algorithm has a linear computational cost at every time step. We illustrate this approach by solving two example non-stationary three-dimensional problems using explicit Euler and Newmark time-stepping scheme: heat equation and linear elasticity equations for a cube. The stability of the simulation is controlled by monitoring the energy of the solution.

Quasilinear poroelasticity: Analysis and hybrid finite element approximation

We consider a system of partial differential equations which models flows through elastic porous media. This system consists of an elasticity equation describing the displacement of an elastic porous matrix and a quasilinear elliptic equation describing the pressure of the saturating fluid (flowing through its pores). In this model, the permeability depends nonlinearly on the dilatation (divergence of the displacement) of the medium. We show that the solution has regularity. We describe the numerical approximation of solutions using a hybrid finite element‐least squares mixed finite element method. Error estimates are obtained through the introduction of an auxiliary linear elasticity equation. Numerical experiments verify the error estimates and validate the proposed poroelasticity model. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1174–1189, 2015

Effective bulk moduli of materials containing stochastically distributed nano‐inhomogeneities with surface stresses

AbstractIn the present contribution, a mathematical model for the investigation of the effective properties of a material with randomly distributed nano‐particles is proposed. The surface effect is introduced via Gurtin‐Murdoch equations describing properties of the matrix/nano‐particle interface. They are added to the system of stochastic differential equations formulated within the framework of linear elasticity. The homogenization problem is reduced to finding a statistically averaged solution of the system of stochastic differential equations. These equations are based on the fundamental equations of linear elasticity, which are coupled with surface/interface elasticity accounting for the presence of surface tension. Using Green's function this system is transformed to a system of statistically non‐linear integral equations. It is solved by the method of conditional moments. Closed‐form expressions are derived for the effective moduli of a composite consisting of a matrix with randomly distributed spherical inhomogeneities. The radius of the nano‐particles is included in the expression for the bulk moduli. As numerical examples, nano‐porous aluminum and nano‐porous gold are investigated assuming that only the influence of the interface effects on the effective bulk modulus is of interest. The dependence of the normalized bulk moduli of nano‐porous aluminum on the pore volume fraction (for certain radii of nano‐pores) are compared to and discussed in the context of other theoretical predictions. The effective Young's modulus of nano‐porous gold as a function of pore radius (for fixed void volume fraction) and the normalized Young's modulus vs. the pore volume fraction for different pore radii are analyzed. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A NONCONFORMING PRIMAL MIXED FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS

Abstract. In this article, we propose and analyze a new nonconformingprimal mixed finite element method for the stationary Stokes equations.The approximation is based on the pseudostress-velocity formulation.The incompressibility condition is used to eliminate the pressure vari-able in terms of trace-free pseudostress. The pressure is then computedfrom a simple post-processing technique. Unique solvability and optimalconvergence are proved. Numerical examples are given to illustrate theperformance of the method. 1. IntroductionBasic mathematical models in fluid and solid mechanics are expressed interms of a set of partial differential equations with the physical unknowns suchas pressure, velocity, stress, and/or an appropriate energy variable. For exam-ple, the original physical equations for incompressible Newtonian flows inducedby the conservation of momentum and the constitutive law are given by thestress-velocity-pressure formulation [18]. The name and many of the originalconcepts for the mixed methods are originated in solid mechanics where it wasdesirable to have simultaneous approximations of certain quantities of inter-est. For the Stokes equations governing flows of incompressible viscous fluids,Galerkin mixed methods based on the pressure-velocity formation are well-understood [19]. The pseudostress-displacement formulation was proposed byArnold and Falk [2] for the equations of linear elasticity which does not requiresymmetric tensors and hence facilitates the use of Raviart-Thomas mixed finiteelements developed for scalar second order elliptic equations. The pseudostress-velocityformulation is then exploited to study first-orderdiv least-squaresfiniteelement methods for the Stokes system by Cai, Lee, and Wang [12].

Artificial conditions for the linear elasticity equations

In this paper, we consider the equations of linear elasticity in an exterior domain. We exhibit artificial boundary conditions on a circle, which lead to a non-coercive second order boundary value problem. In the particular case of an axisymmetric geometry, explicit computations can be performed in Fourier series proving the well-posedness except for a countable set of parameters. A perturbation argument allows to consider near-circular domains. We complete the analysis by some numerical simulations.

Numerical analysis of leakage of elastomeric seals for reciprocating circular motion

A cambered elastomeric seal used in a novel universal joint for reciprocating circular motion is investigated. Then based on the linear elastic Hooke equations and the Reynolds equation, mathematical models of the contact pressure, film thickness and leakage with the method of variable substitution are established. The numerical analysis and simulation are performed based on the modified inverse hydrodynamic (IH) method and MATLAB numerical method. Besides, effects of various parameters on the sealing performance are investigated systematically. The simulation results verify the effectiveness and feasibility of proposed mathematical model and numerical algorithm. The results also lay the theoretical basis for the structure design and performance analysis of the seal assembly.

A partitioned scheme for fluid–composite structure interaction problems

We present a benchmark problem and a loosely-coupled partitioned scheme for fluid–structure interaction with composite structures. The benchmark problem consists of an incompressible, viscous fluid interacting with a structure composed of two layers: a thin elastic layer with mass which is in contact with the fluid and modeled by the Koiter membrane/shell equations, and a thick elastic layer with mass modeled by the equations of linear elasticity. An efficient, modular, partitioned operator-splitting scheme is proposed to simulate solutions to the coupled, nonlinear FSI problem, without the need for sub-iterations at every time-step. An energy estimate associated with unconditional stability is derived for the fully nonlinear FSI problem defined on moving domains. Two instructive numerical benchmark problems are presented to test the performance of numerical FSI schemes involving composite structures. It is shown numerically that the proposed scheme is at least first-order accurate both in time and space. This work reveals a new physical property of FSI problems involving thin interfaces with mass: the inertia of the thin fluid–structure interface regularizes solutions to the full FSI problem.

A Rigorous Derivation of the Equations for the Clamped Biot-Kirchhoff-Love Poroelastic Plate

In this paper we investigate the limit behavior of the solution to quasi-static Biot's equations in thin poroelastic plates as the thickness tends to zero. We choose Terzaghi's time corresponding to the plate thickness and obtain the strong convergence of the three-dimensional solid displacement, fluid pressure and total poroelastic stress to the solution of the new class of plate equations. In the new equations the in-plane stretching is described by the 2D Navier's linear elasticity equations, with elastic moduli depending on Gassmann's and Biot's coefficients. The bending equation is coupled with the pressure equation and it contains the bending moment due to the variation in pore pressure across the plate thickness. The pressure equation is parabolic only in the vertical direction. As additional terms it contains the time derivative of the in-plane Laplacean of the vertical deflection of the plate and of the the elastic in-plane compression term.

Compliant topology optimization for planar passive flap micro valve.

This paper reports the compliant topology optimization for planar passive flap micro valve considering fluid-structure interaction with a monolithic approach. Although flap valve type check valve is easy to manufacture and use for the applications for Bio/Nano/MEMS, its structural optimization has been seldom conducted so far. The size of the Bio/Nano/MEMS devices becomes smaller and the simple straight type micro valve structure is required to be optimized considering fluid speed. To address this optimization problem, the structural topology optimization scheme which designs optimal topologies is applied for a flap type check valve structure. To consider the coupling effects of fluid domain and structural domain, the monolithic finite element approach is employed. In the new analysis approach, solid domain is simulated by introducing the inverse permeability in the Navier-Stokes equation and the fluid stress filter in the linear elasticity equation. Also it is a new idea that fluid domain is simulated by finite elements with a weak Young's modulus in the linear elasticity equation. The mutual couplings between fluid and structure are considered by the introduction of the deformation tensor which is one of the basic concepts of the continuum mechanism. By distributing material properties inside a design domain for compliant flap, optimal flap structures can be constructed with different fluid speeds. By investigating the optimal layouts of several passive flap designs, we prove that the structural topology optimization can provide optimal layouts for Bio, Nano, and MEMS applications.

An adaptive finite element procedure for fully-coupled point contact elastohydrodynamic lubrication problems

This paper presents an automatic locally adaptive finite element solver for the fully-coupled EHL point contact problems. The proposed algorithm uses a posteriori error estimation in the stress in order to control adaptivity in both the elasticity and lubrication domains. The implementation is based on the fact that the solution of the linear elasticity equation exhibits large variations close to the fluid domain on which the Reynolds equation is solved. Thus the local refinement in such region not only improves the accuracy of the elastic deformation solution significantly but also yield an improved accuracy in the pressure profile due to increase in the spatial resolution of fluid domain. Thus, the improved traction boundary conditions lead to even better approximation of the elastic deformation. Hence, a simple and an effective way to develop an adaptive procedure for the fully-coupled EHL problem is to apply the local refinement to the linear elasticity mesh. The proposed algorithm also seeks to improve the quality of refined meshes to ensure the best overall accuracy. It is shown that the adaptive procedure effectively refines the elements in the region(s) showing the largest local error in their solution, and reduces the overall error with optimal computational cost for a variety of EHL cases. Specifically, the computational cost of proposed adaptive algorithm is shown to be linear with respect to problem size as the number of refinement levels grows.

Stiffness Constants of Homogeneous, Anisotropic, Prismatic Beams

This paper presents a complete set of analytical expressions for the stiffness constants of a generalized Euler–Bernoulli beam theory for homogeneous, anisotropic, prismatic beams with arbitrary cross-sectional shapes. These expressions are extracted from exact solutions of the linear equations of three-dimensional elasticity for the cases of loading by axial forces, torques, and bending moments about two orthogonal directions. Closed-form expressions are derived for the extensional stiffness and the extension-related coupling terms. Expressions for the remaining stiffness constants are derived in terms of the torsional stiffness: the expression of which is in terms of a function that needs to be obtained. The resulting expressions reveal both similarities and differences from its isotropic and orthotropic counterparts. For elliptical, anisotropic cross sections and rectangular, orthotropic cross sections, all stiffness constants are known in closed form. These closed-form expressions constitute a standard with which the ability of two-dimensional beam cross-sectional analyses to model material anisotropy may be assessed. The calculated stiffness constants, from one such cross-sectional analysis, are successfully validated in this manner.

Crista egregia: a geometrical model of the crista ampullaris, a sensory surface that detects head rotations.

The crista ampullaris is the epithelium at the end of the semicircular canals in the inner ear of vertebrates, which contains the sensory cells involved in the transduction of the rotational head movements into neuronal activity. The crista surface has the form of a saddle, or a pair of saddles separated by a crux, depending on the species and the canal considered. In birds, it was described as a catenoid by Landolt et al. (J Comp Neurol 159(2):257-287, doi: 10.1002/cne.901590207 ,1972). In the present work, we establish that this particular form results from principles of invariance maximization and energy minimization. The formulation of the invariance principle was inspired by Takumida (Biol Sci Space 15(4):356-358,2001). More precisely, we suppose that in functional conditions, the equations of linear elasticity are valid, and we assume that in a certain domain of the cupula, in proximity of the crista surface, (1) the stress tensor of the deformed cupula is invariant under the gradient of the pressure, (2) the dissipation of energy is minimum. Then, we deduce that in this domain the crista surface is a minimal surface and that it must be either a planar, or helicoidal Scherk surface, or a piece of catenoid, which is the unique minimal surface of revolution. If we add the hypothesis that the direction of invariance of the stress tensor is unique and that a bilateral symmetry of the crista exists, only the catenoid subsists. This finding has important consequences for further functional modeling of the role of the vestibular system in head motion detection and spatial orientation.

A semi‐local spectral/hp element solver for linear elasticity problems

SUMMARYWe develop an efficient semi‐local method for speeding up the solution of linear systems arising in spectral/hp element discretization of the linear elasticity equations. The main idea is to approximate the element‐wise residual distribution with a localization operator we introduce in this paper, and subsequently solve the local linear system. Additionally, we decouple the three directions of displacement in the localization operator, hence enabling the use of an efficient low energy preconditioner for the conjugate gradient solver. This approach is effective for both nodal and modal bases in the spectral/hp element method, but here, we focus on the modal hierarchical basis. In numerical tests, we verify that there is no loss of accuracy in the semi‐local method, and we obtain good parallel scalability and substantial speed‐up compared to the original formulation. In particular, our tests include both structure‐only and fluid‐structure interaction problems, with the latter modeling a 3D patient‐specific brain aneurysm. Copyright © 2014 John Wiley & Sons, Ltd.