Equations Of Linear Elasticity Research Articles

Quantitative osteoporosis diagnosis of porous cancellous bone using poroelastodynamic modal analysis.

Osteoporosis is a skeletal disease characterized by reduced bone mass and microarchitectural deterioration, leading to increased fragility. This study presents a novel three-dimensional poroelastodynamic model for analyzing cancellous bone free vibration responses. The model incorporates the Navier-Stokes equations of linear elasticity and the Biot theory of porous media, allowing the investigation of osteoporosis-related changes. The analysis considers parameters like porosity, density, elasticity, Poisson ratio, and viscosity of bone marrow within the porous medium. Our findings indicate that natural frequencies of cancellous bone play a crucial role in osteoporosis prediction. By incorporating experimental data from 12 mouse femurs, we unveil insights into osteoporosis prediction. Increased porosity reduces bone stiffness, lowering natural frequencies. However, it also increases bone mass loss relative to stiffness, leading to higher frequencies. Therefore, the natural frequencies of osteoporotic bone are always higher than the natural frequencies of normal bone. Additionally, an increase in bone marrow within the pores, while increasing damping effects, also increases natural frequencies, which is another indication of osteoporosis growth in bone. The presence of bone marrow within the pores further influences natural frequencies, providing additional insights into osteoporosis growth. Thinner and smaller bones are found to be more susceptible to osteoporosis compared to larger and bigger bones due to their higher natural frequencies at equivalent porosity levels.

Analysis‐aware defeaturing of complex geometries with Neumann features

AbstractLocal modifications of a computational domain are often performed in order to simplify the meshing process and to reduce computational costs and memory requirements. However, removing geometrical features of a domain often introduces a non‐negligible error in the solution of a differential problem in which it is defined. In this work, we extend the results from Buffa et al. (Math Models Methods Appl Sci. 2022; 32(02):359–402.) by studying the case of domains containing an arbitrary number of distinct Neumann features, and by performing an analysis on Poisson's, linear elasticity, and Stokes' equations. We introduce a simple, computationally cheap, reliable, and efficient a posteriori estimator of the geometrical defeaturing error. Moreover, we also introduce a geometric refinement strategy that accounts for the defeaturing error: starting from a fully defeatured geometry, the algorithm determines at each iteration step which features need to be added to the geometrical model to reduce the defeaturing error. These important features are then added to the (partially) defeatured geometrical model at the next iteration, until the solution attains a prescribed accuracy. A wide range of two‐ and three‐dimensional numerical experiments are finally reported to illustrate this work.

Isogeometric Tearing and Interconnecting solvers for linear elasticity in multi-patch Isogeometric Analysis with theory for two dimensional domains

We consider the linear elasticity equation, discretized with multi-patch Isogeometric Analysis. A standard discretization error analysis is based on Korn’s inequality, which degrades for certain geometries, such as long and thin cantilevers. This phenomenon is known as geometry locking. We observe that high-order methods, like Isogeometric Analysis is beneficial in such a setting. The main focus of this paper is the construction and analysis of a domain decomposition solver, namely an Isogeometric Tearing and Interconnecting (IETI) solver, where we prove that the convergence behavior does not depend on the constant of Korn’s inequality for the overall domain, but only on the corresponding constants for the individual patches. Moreover, our analysis is explicit in the choice of the spline degree. Numerical experiments are provided which demonstrates the efficiency of the proposed solver.

Deformation of an elastic second gradient spherical body under equatorial line density of dead forces

We consider deformations of an elastic body having initially a spherical shape. Assumed deformation energy depends on the first and second gradient of displacements. We apply an equatorial line density of dead loads, that are forces per unit line length directed in radial direction and applied along the equator of the sphere. We restrict ourselves our analysis to the case of linearized second strain gradient isotropic elasticity (for which the more general energy was determined by Mindlin) with only one characteristic length. Differently to what happens in first gradient continua, i.e. in classic linear elasticity, we show that for the particular class second gradient continua considered here these forces do not determine infinite displacements in the direction of applied dead line forces. Instead, using a series method for the solution of the considered boundary-value problem, we demonstrate that the displacements are finite. So in the deformed configuration there is not the formation of an edge at the material points where the forces are applied. Further investigations are therefore needed for establishing if this elastic-regime edge formation is made possible: (I) either in the case of more general linear elastic constitutive equations or (II) only when large deformations are considered or (III) if non-elastic phenomena are involved.

Deep Ritz method with adaptive quadrature for linear elasticity

In this paper, we study the deep Ritz method for solving the linear elasticity equation from a numerical analysis perspective. A modified Ritz formulation using the H1/2(ΓD) norm is introduced and analyzed for linear elasticity equation in order to deal with the (essential) Dirichlet boundary condition. We show that the resulting deep Ritz method provides the best approximation among the set of deep neural network (DNN) functions with respect to the “energy” norm. Furthermore, we demonstrate that the total error of the deep Ritz simulation is bounded by the sum of the network approximation error and the numerical integration error, disregarding the algebraic error. To effectively control the numerical integration error, we propose an adaptive quadrature-based numerical integration technique with a residual-based local error indicator. This approach enables efficient approximation of the modified energy functional. Through numerical experiments involving smooth and singular problems, as well as problems with stress concentration, we validate the effectiveness and efficiency of the proposed deep Ritz method with adaptive quadrature.

Physics-informed neural networks for mesh deformation with exact boundary enforcement

In this work, we have applied physics-informed neural networks (PINN) for solving mesh deformation problems. We used the collocation PINN method to capture the new positions of the vertex nodes while preserving the connectivity information. We use linear elasticity equations for mesh deformation. To prevent vertex collisions or edge overlap, the mesh movement in this work is conducted in steps with relatively small movements. For moving boundary problems, the exact position of the boundary is essential for having an accurate solution. However, PINNs are frequently unable to satisfy Dirichlet boundary conditions exactly. To overcome this issue, we have used hard boundary condition enforcement to automatically satisfy Dirichlet boundary conditions. Specifically, we first trained a PINN with soft boundary conditions to obtain a particular solution. Then, this solution was tuned with exact boundary positions and a proper distance function by using a new PINN considering only the equation residual. To assess the accuracy of our approach, we used the classical translation and rotation tests and compared them with a proper mesh quality metric considering the change in the element area and shape. The results show the accuracy of this approach is comparable with that of finite element solutions. We also solved different moving boundary problems, resembling commonly used fluid–structure interaction problems. This work provides insight into using PINN for mesh-deformation problems without needing a discretization scheme with reasonable accuracy.

Structural optimisation of H-Rotor wind turbine blade based on one-way Fluid Structure Interaction approach

The paper aims to present structural topology optimisation of the structural topology of the H-Rotor wind turbine combined with the one-way Fluid Structure Interaction (FSI) approach. The developed methodology couples the set of Unsteady Reynolds Averaged Navier–Stokes Equations (URANS) with the steady state linear elasticity equations and density based topology optimisation method. This approach allowed us to reduce the mass of the blades with respect to the maximum stress and deformation limits. The results obtained provide detailed information on the unsteady flow fields around the operating wind turbine, as well as optimised topology of the blade interior without affecting the external aerodynamic profile. The load of pressure profile from the CFD simulation was implemented at the point of rotation at which the force values observed on the blade surface were the highest. The centrifugal force was considered in the structural model. Computations were carried out for wind speed equal to 30 m/s and rotational velocity equal to 90 rad/s. The turbine operating parameters were chosen to represent extreme operating conditions. The results obtained showed that the blade mass, made of the same material, can be easily reduced by around 60% with the reduction of the observed stress values and deformation.

"THE ELASTIC CONTACT PROBLEM INVOLVING A BILATERALLY LOADED THIN STRIP "

"For the solution of the elastic contact problem, it is generally assumed that the contacting solids can be assimilated to elastic half-spaces. This assumption is reasonable when the contact area is small compared to the dimensions of the contacting solids, and when the contact stresses are restricted to a small vicinity of the initial point of contact, without reaching the boundaries of the solids. These conditions are not met when a thin strip is bilaterally and symmetrically compressed between two punches, as the dimensions of the contact area might be of the same order of magnitude as the strip thickness. Moreover, the stresses induced in the strip will reach the boundary without a significant decay in intensity. Consequently, in this contact scenario, the classical solutions for a point force acting on the boundary of the half-space cannot be directly applied. The starting point for the problem solution is a modified Boussinesq-type solution for the thin elastic layer, expressing the displacement and stress fields induced in an elastic strip by two opposed normal forces perpendicular to the layer boundaries. To this end, supplementary displacements are added to the half-space solution to satisfy both geometric and loading symmetry, as well as the linear elasticity equations. Superposition principle is then applied, resulting in a contact model for an elastic strip compressed between two identical indenters with aligned axes. An algorithm for the contact of solids that can be assimilated to elastic half-spaces is modified and applied to the contact involving a thin strip. To this end, the required influence coefficients for displacements are derived. A calculation example involving a thin strip compressed between two spheres is presented, and a comparison with the half-space solution is performed. "

Kernel free boundary integral method for 3D incompressible flow and linear elasticity equations on irregular domains

A second-order accurate kernel-free boundary integral method is presented for Stokes flow and linear elasticity boundary value problems on three-dimensional irregular domains. It solves equations in the framework of boundary integral equations, whose corresponding discrete forms are well-conditioned and solved by the GMRES method. A notable feature of this approach is that the boundary or volume integrals encountered in BIEs are indirectly evaluated by a Cartesian grid-based method, which includes discretizing corresponding simple interface problems with a MAC scheme, correcting discrete linear systems to reduce large local truncation errors near the interface, solving the modified system by a CG method together with an FFT-based Poisson solver. No extra work or special quadratures are required to deal with singular or hyper-singular boundary integrals and the dependence on the analytical expressions of Green’s functions for the integral kernels is completely eliminated. Numerical results are given to demonstrate the efficiency and accuracy of the Cartesian grid-based method.

Finite element modeling and simulation of mixed elastohydrodynamic lubrication for the finite line contact under tilting loads

Misalignment is a prevalent problem in rotor machinery. This paper proposes a numerical analysis for the mixed EHL of finite-line contacts considering tilting loads based on FEM. The Reynolds and linear elasticity equations, and Greenwood-Tripp contact model are coupled in an iterative process to solve the fluid-solid/solid-solid interaction under the compound external force and moment. Lubrication characteristics at both ends of the cylinder are analyzed and an asymmetric profile design scheme is presented to improve the lubrication performance at the cylinder's lower end. It is found that proper profile parameters can reduce EHL pressure by 40%, increasing the minimum oil film thickness by up to 100%, and thus reduce asperity contact pressure by up to 50%.

Fractional-step finite difference schemes for incompressible elasticity on overset grids

Efficient finite-difference schemes for the numerical solution of the time-dependent equations of incompressible linear elasticity on complex geometry are presented. The schemes are second-order accurate in space and time, and solve the equations in displacement-pressure form. The algorithms use a fractional-step approach in which the time-step of the displacements is performed separately from the solution of a Poisson problem to update the pressure. Complex geometry is treated with curvilinear overset grids. Compatibility boundary conditions and an upwind dissipation for wave equations in second-order form are included to ensure stability for overset grids, and for the difficult case of traction (free-surface) boundary conditions. A divergence damping term is added to keep the dilatation small. The stability of the schemes is studied with GKS mode analysis. Exact eigen-mode solutions are obtained for several problems involving rectangular, cylindrical and spherical configurations, and these are valuable as benchmark problems. The accuracy and stability of the approach in two and three space dimensions is illustrated by comparing the numerical solutions with the exact solutions of the benchmark problems.

Numerical simulations of the wake and deformations of a flexible rotor blade in a turbulent flow

We present, for the first time, the mean deflection evolution of a flexible rotor blade using a coupled model based on Navier–Stokes equations, for the fluid flow, and linear elasticity equations for the blade deformation. Three turbulence models are tested to reach Reynolds numbers as high as 8 104. The absolute tip speed ratios are in the range [0,25]. The numerical results are validated by comparisons with available tip displacements from experiments. For the parameter ranges, above mentioned, the elastic behavior of the flexible rotor is characterized, and the vorticity field is compared with results obtained for a rigid rotor. The effects of the pitch, the tip speed ratio (or frequency), and its sign on the blade deformation are reported. Typically, the blade deforms in the downstream direction, and it is shown that this deformation is a non-monotonic function of the rotation frequency and the pitch angle. Furthermore, it is found that, for particular values of the frequency and pitch angle, the blade is subject to deformations in the upstream direction. It is shown also that the flexible rotor could develop a vortex ring state, but not the rigid one, under the same conditions. It is found that there is a supercritical frequency associated with the apparition of this vortex ring state and this frequency occurs for negative pitches only, for the considered blade. The vorticity field revealed, as well, that the tip vortex changes sign with that of the blade deflection. Finally, we present the effect of the pitch and frequency on the twist angle of the blade and characterize its evolution along the span.

A mixed virtual element method for nearly incompressible linear elasticity equations

A mixed virtual element method for nearly incompressible linear elasticity equations

Spectral and Evolution Analysis of Composite Elastic Plates with High Contrast

We analyse the behaviour of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, we calculate the limit resolvent and analyse the associated limit spectrum and effective evolution equations. We obtain our results under various asymptotic relations between the size of the soft inclusions (equivalently, the period) and the plate thickness as well as under various scaling combinations between the contrast, spectrum, and time. In particular, we demonstrate significant qualitative differences between the asymptotic models obtained in different regimes.

A new lattice Boltzmann scheme for linear elastic solids: periodic problems

We propose a new second-order accurate lattice Boltzmann scheme that solves the quasi-static equations of linear elasticity in two dimensions. In contrast to previous works, our formulation solves for a single distribution function with a standard velocity set and avoids any recourse to finite difference approximations. As a result, all computational benefits of the lattice Boltzmann method can be used to full capacity. The novel scheme is systematically derived using the asymptotic expansion technique and a detailed analysis of the leading-order error behavior is provided. As demonstrated by a linear stability analysis, the method is stable for a very large range of Poisson’s ratios. We consider periodic problems to focus on the governing equations and rule out the influence of boundary conditions. The analytical derivations are verified by numerical experiments and convergence studies.

A linear two-dimensional mathematical model for thin two-layer plates with partial shear interaction, with a view towards application to laminated glass

This work presents a consistent derivation, from three-dimensional linear elasticity, of a two-dimensional mathematical model describing the bending and in-plane stretching behaviours, under a general system of quasi-static distributed loads and prescribed support displacements, of thin two-layer plates with partial shear interaction. This layerwise model is specifically tailored for the requirements posed by the analysis of laminated glass plates commonly used in building structures (consisting of two thin glass layers bonded together by an adhesive interlayer). Our approach, based on Podio-Guidugli’s method of internal constraints, avoids mutually contradictory assumptions (not uncommon in the literature on structural mechanics) and yields a complete two-dimensional characterisation of displacement, strain and stress fields that exactly satisfy the field equations of three-dimensional linear elasticity and the boundary conditions at the end faces. The choice of generalised variables is designed to bring to light the following fundamental conclusion and physical insight: the resulting two-dimensional boundary value problem is a combination of the equations of Kirchhoff and Mindlin plates (with specified rigidities).Two examples illustrate the application of the proposed model: (i) the cylindrical bending of plate strips and (ii) a family of problems with Navier-type analytical solution. The solutions exhibit continuity across the whole range of zero, partial and full interaction between the layers. Moreover, the Navier-type solutions are consistently in close agreement with the results of three-dimensional finite element analyses. On the contrary, analogous results previously reported in the literature exhibit considerable deviations. An explanation for these discrepancies is discussed in detail.

Local multiscale model reduction using discontinuous Galerkin coupling for elasticity problems

In this paper, we consider the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with discontinuous Galerkin (DG) coupling for the linear elasticity equations in highly heterogeneous and high contrast media. We will introduce the construction of a DG version of the CEM-GMsFEM, such as auxiliary basis functions and offline basis functions. The DG version of the method offers some advantages such as flexibility in coarse grid construction and sparsity of resulting discrete systems. Moreover, to our best knowledge, this is the first time where the proof of the convergence of the CEM-GMsFEM in the DG form is given. Some numerical examples will be presented to illustrate the performance of the method.

Shape optimization for composite materials in linear elasticity

This article is devoted to the optimal design of the microstructure in composite materials, which are governed by the equations of linear elasticity. To this end, we combine homogenization with shape optimization. In particular, we determine the sensitivity of the homogenized coefficients of the elasticity tensor with respect to the shape of the periodic microstructure also in case of spatially varying material coefficients. We compute the respective Hadamard shape gradient and demonstrate the applicability and feasibility of our approach by numerical experiments for different problem settings.

Asymptotic analysis of 3D dynamic equations in linear elasticity for a thin layer resting on a Winkler foundation

Abstract A 3D dynamic problem for a thin elastic layer resting on a Winkler foundation is considered. The stiffness of the layer is assumed to be much greater than that of the foundation in order to allow low-frequency bending motion. The leading long-wave approximation valid outside the vicinity of the cut-off frequency is derived. It is identical to the classical Kirchhoff plate theory. A novel near cut-off 2D approximation is also established. It involves both bending and extension motions which are not separated from each other due to the effect of the foundation. The associated dispersion relation appears to be non-uniform over the small wavenumber domain.

The MHM Method for Linear Elasticity on Polytopal Meshes

Abstract The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient conditions on the fine-scale interpolations such that the MHM method is well-posed and optimally convergent under local regularity conditions. Also, a multi-level error analysis demonstrates that the MHM method achieves convergence without refining the coarse partition. The upshot is that the Poincaré and Korn’s inequalities do not degenerate, and then the convergence arises on general meshes. Two- and three-dimensional numerical tests assess theoretical results and verify the robustness of the method on a multi-layer media case.