Equations Of Linear Elasticity Research Articles

Investigation of Seismic Stability of High-Rising Buildings Using Grid-Characteristic Method

The purpose of this work was the numerical simulation of dynamic wave processes in heterogeneous media with the subsequent evaluation of their influence on various structures. The effect of earthquakes on ground buildings and underground structures was studied. In the work, we used the grid-characteristic method, which allows taking into account the physics of arising wave processes. The use of hierarchical grid systems allows the calculation of seismic waves directly from the earthquake source to ground-based buildings and underground structures. The work investigated the destruction of high-rise buildings under the influence of seismic loads. The emerging areas of fractures in various structures were studied. The effect of the earthquake on the underground structure was considered. The influence of the depth of the earthquake source was analyzed. Images of wave processes in structures and places of destruction were obtained. Note that we do not calculate the damage caused by falling of some collapsed parts of the building to others, but we calculate only damage caused directly by seismic activity. This type of damage is of the greatest interest, since it is necessary to avoid precisely this primary destruction. Also we investigate the nature of this type of damage, e.g, internal, near the surface, causing collapse, asymmetric. The novelty of the approach used also lies in the fact that we solve a linear elastic wave equation with constant coefficients in various subdomains of the integration domain, between which contact conditions are established. And the size and shape of these subdomains vary over time and depend on the solution. That is, the complete boundary problem of the elastic wave equation is nonlinear.

Preconditioning the coarse problem of BDDC methods ‐ three-level, algebraic multigrid, and vertex-based preconditioners

A fair comparison of three Balancing Domain Decomposition by Constraints (BDDC) methods with an approximate coarse space solver is attempted for the first time. The comparison is made for a BDDC method with an algebraic multigrid preconditioner for the coarse problem, a three-level BDDC method, and a BDDC method with a vertex-based coarse preconditioner which was recently introduced by Clark Dohrmann, Kendall Pierson, and Olof Widlund. For the first time, all methods are presented and discussed in a common framework. Condition number bounds are provided for all approaches. All methods are implemented in a common highly parallel scalable BDDC software package based on PETSc, to allow for a fair comparison. Numerical results showing the parallel scalability are presented for the equations of linear elasticity. For the first time, this includes parallel scalability tests for the vertex-based approximate BDDC method.

Nonconforming mixed finite elements for linear elasticity on simplicial grids

This paper introduces a new family of nonconforming mixed finite elements for solving the linear elasticity equations on simplicial grids. Besides, this paper describes the construction of the lowest order basis functions. The construction only involves simple computations due to the new explicit stress shape function spaces and the procedure applies for high order cases. Numerical experiments for four benchmark problems in mechanics indicate the robust locking‐free behavior and show that the lowest order nonconforming mixed method leads to smaller stress errors than the first and second order standard Galerkin methods for the nearly incompressible case.

Vibration of thermo lemv composite multilayered hollow pipes

The present paper is concerned the effects of thermo elastic waves in a composite multilayered hollow pipes which contain inner and outer zinc layers bonded by Linear Elastic Material with Voids (LEMV) is considered. The equation of movement is derived by means of the constitutive equations of linear thermo elasticity. The equation of movement and heat conduction models are decoupled by the displacement of potentials which are constructed based on equilibrium equations of elasticity. The dispersion equation are acquired by means of traction free boundary conditions and are numerically analyzed for Zinc material. The enumerated frequency, phase velocity are presented graphically for LEMV and CFRP layers.

Spatial Poisson processes for fatigue crack initiation

In this work we propose a stochastic model for estimating the occurrence of crack initiations on the surface of metallic specimens in fatigue problems that can be applied to a general class of geometries. The stochastic model is based on spatial Poisson processes with intensity function that combines stress-life (S–N) curves with averaged effective stress, σeffΔ(x), which is computed after solving numerically the linear elasticity equations on the specimen domains using finite element methods. Here, Δ is a parameter that characterizes the size of the neighbors covering the domain boundary. The averaged effective stress, parameterized by Δ, maps the stress tensor to a scalar field upon the specimen domain. Data from fatigue experiments on notched and unnotched sheet specimens of 75S-T6 aluminum alloys are used to calibrate the model parameters for the individual data sets and their combination. Bayesian and classical approaches are applied to estimate the survival-probability function for any specimen tested under a prescribed fatigue experimental setup. Our proposed model can predict the initiation of cracks in specimens made from the same material with new geometries.

Scalloping and Stress Concentration in DRIE-Manufactured Comb-Drives

In the last decades, microelectromechanical systems have been increasing their number of degrees of freedom and their structural complexity. Hence, most recently designed MEMSs have required higher mobility than in the past and higher structural strength and stability. In some applications, device thickness increased up to the order of tens (or hundred) of microns, which nowadays can be easily obtained by means of DRIE Bosch process. Unfortunately, scalloping introduces stress concentration regions in some parts of the structure. Stress concentration is a dangerous source of strength loss for the whole structure and for comb-drives actuators which may suffer from side pull-in. This paper presents an analytical approach to characterize stress concentrations in DRIE micro-machined MEMS. The method is based on the linear elasticity equations, the de Saint-Venant Principle, and the boundary value problem for the case of a torsional state of the beam. The results obtained by means of this theoretical method are then compared with those obtained by using two other methods: one based on finite difference discretization of the equations, and one based on finite element analysis (FEA). Finally, the new theoretical approach yields results which are in accordance with the known value of the stress concentration factor for asymptotically null radius notches.

Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine

In this paper we discuss a new and very efficient implementation of high order accurate arbitrary high order schemes using derivatives discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear systems of hyperbolic partial differential equations. ADER-DG schemes are by construction communication-avoiding and cache-blocking, and are furthermore very well-suited for vectorization, and so they appear to be a good candidate for the future generation of exascale supercomputers. We introduce the numerical algorithm and show some applications to a set of hyperbolic equations with increasing levels of complexity, ranging from the compressible Euler equations over the equations of linear elasticity and the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of general relativity. We present strong scaling results of the new ADER-DG schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems. We also provide a detailed performance comparison with traditional Runge-Kutta DG schemes.

A symplectic procedure for two-dimensional coupled elastic wave equations using radial basis functions interpolation

This study proposes a meshless symplectic algorithm for two-dimensional coupled elastic waves, which satisfy the equations of linear isotropic elasticity on a rectangular domain with appropriate initial and boundary conditions. This method is based on a combination of radial basis functions (RBFs) collocation and geometric integrators. The method involves two steps, first the spatial discretization by using RBFs interpolation transforms the seismic waves into a finite-dimensional Hamiltonian system, then evolves the semi-discretized system by using symplectic integrators. Furthermore, the conservations of energy and symplectic structure are investigated. The truncation error and global error of the method are estimated. Numerical experiments show that the scheme is efficient, easy to implement with the scattered knots and possesses a good behavior in the long time integration.

Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media

In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, generalizing the classical Biot equations. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the use of separate simulators. It is not met by the classical Newton method. We propose three different linearization schemes incorporating the fixed-stress splitting scheme, coupled with an L-scheme, Modified Picard and Newton linearization of the flow equations. All schemes allow the efficient and robust decoupling of mechanics and flow equations. In particular, the simplest scheme, the Fixed-Stress-L-scheme, employs solely constant diagonal stabilization, has low cost per iteration, and is very robust. Under mild, physical assumptions, it is theoretically shown to be a contraction. Due to possible break-down or slow convergence of all considered splitting schemes, Anderson acceleration is applied as post-processing. Based on a special case, we justify theoretically the general ability of the Anderson acceleration to effectively accelerate convergence and stabilize the underlying scheme, allowing even non-contractive fixed-point iterations to converge. To our knowledge, this is the first theoretical indication of this kind. Theoretical findings are confirmed by numerical results. In particular, Anderson acceleration has been demonstrated to be very effective for the considered Picard-type methods. Finally, the Fixed-Stress-Newton scheme combined with Anderson acceleration shows the best performance among the splitting schemes.

Influence of Rotation on Transversely Isotropic Piezoelectric Rod Coated with a Thin Film

In this paper, the influence of rotation on axisymmetric waves of a piezoelectric rod coated with a thin film is studied using constitutive form linear theory elasticity and piezo-electric equations. Potential functions are introduced to uncouple the equations of motion in radial and axial directions. The surface area of the rod is coated by a perfectly conducting material. The frequency equations are obtained for longitudinal and flexural modes of vibration and are studied numerically for PZT-4 ceramics. The computed non-dimensional frequency, phase velocity, relative frequency shift, electromechanical coupling and electric displacement are presented in the form of dispersion curves. This type of study is important in the construction of rotating sensors and gyroscope.

Thermo-elasto-plastic analysis of geosynthetic clay liners exposed to thermal dehydration

Geosynthetic clay liners (GCLs) are designed to minimise contaminant migration into subsoils in landfills and other waste containment systems. The so-called mixture theory has typically been chosen as a basis for numerical models describing the complicated behaviour of GCLs subjected to thermo-hydro-mechanical changes. Despite progress in this area in the past decade, some aspects of the response of GCLs have not been captured by these simulations. For example, a typical assumption of these analyses is that the mechanical response to thermal loads of GCLs can be described by linear or non-linear elastic constitutive equations. However, there is evidence that in many instances plastic deformation cannot be avoided. The primary aim of this study is to analyse the hydration and dehydration behaviour of GCLs within a thermo-elasto-plastic framework. First, a thermo-elasto-plastic constitutive model is introduced and numerically implemented. Next, the results of numerical simulations of the performance of GCLs are presented and discussed and the effects of deviations from elastic behaviour are identified. Finally, a parametric study is conducted to identify the influence of key parameters in the constitutive model on the horizontal stress in the GCL’s bentonite during restrained shrinkage driven by thermal dehydration.

Pseudo-spectral least squares method for linear elasticity

The first order system least squares Legendre and Chebyshev spectral method for two dimensional space linear elasticity is investigated. The drilling rotation is defined as a new variable and the linear elasticity equation is supplemented with an auxiliary equation. The weighted L2-norm least squares principle is applied to a stress–displacement–rotation. It is shown that the homogeneous least squares functional is equivalent to weighted H1-norm like for stress and weighted H1-norm for displacement and rotation. This weighted H1-norm equivalence is λ-uniform. Spectral convergence for both Legendre and Chebyshev approaches are given along with some numerical experiments. The generalization for three dimensional spaces is also provided.

Space-time L 2 estimates, regularity and almost global existence for elastic waves

Abstract In this paper, we first establish a kind of weighted space-time L 2 {L^{2}} estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L 2 L^{2} estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in H 3 × H 2 {H^{3}\times H^{2}} .

An energy-based discontinuous Galerkin discretization of the elastic wave equation in second order form

We present an application of our general formulation (Appelö and Hagstrom (2015) [12]) to construct energy based, arbitrary order accurate, discontinuous Galerkin spatial discretizations of the linear elastic wave equation. The resulting methods are stable and, depending on the choice of numerical flux, conserve or dissipate the elastic energy. The performance of the method is demonstrated for problems with manufactured and exact solutions. Applications to more realistic problems are also presented. Implementations of the methods are freely available at Appelö and Hagstrom (2015) [19].

Arbitrary high order accurate space–time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity

In this paper we propose a new high order accurate space–time discontinuous Galerkin (DG) finite element scheme for the solution of the linear elastic wave equations in first order velocity-stress formulation in two and three-space dimensions on staggered unstructured triangular and tetrahedral meshes. The method reaches arbitrary high order of accuracy in both space and time via the use of space–time basis and test functions. Within the staggered mesh formulation, we define the discrete velocity field in the control volumes of a primary mesh, while the discrete stress tensor is defined on a face-based staggered dual mesh. The space–time DG formulation leads to an implicit scheme that requires the solution of a linear system for the unknown degrees of freedom at the new time level. The number of unknowns is reduced at the aid of the Schur complement, so that in the end only a linear system for the degrees of freedom of the velocity field needs to be solved, rather than a system that involves both stress and velocity. Thanks to the use of a spatially staggered mesh, the stencil of the final velocity system involves only the element and its direct neighbors and the linear system can be efficiently solved via matrix-free iterative methods. Despite the necessity to solve a linear system, the numerical scheme is still computationally efficient. The chosen discretization and the linear nature of the governing PDE system lead to an unconditionally stable scheme, which allows large time steps even for low quality meshes that contain so-called sliver elements. The fully discrete staggered space–time DG method is proven to be energy stable for any order of accuracy, for any mesh and for any time step size. For the particular case of a simple Crank–Nicolson time discretization and homogeneous material, the final velocity system can be proven to be symmetric and positive definite and in this case the scheme is also exactly energy preserving. The new scheme is applied to several test problems in two and three space dimensions, providing also a comparison with high order explicit ADER-DG schemes.

Analytical elastic models of finite cylindrical and truncated spherical inclusions

We develop a new technique for finding elastic fields for axisymmetric dilatational inclusions (DIs) in the forms of finite cylinder and truncated sphere, when the DIs and surrounding infinite matrix have the same isotropic elastic moduli. DIs are built of circular dilatational disks distributed continuously along the axis of symmetry. Total displacements of DIs are found by integration of the displacements of a dilatational disk. Then, using the linear elasticity equations, the elastic fields of cylindrical and truncated spherical DIs are derived and written via compact easy-to-read and easy-to-calculate Lipschitz-Hankel integrals and Lur'e series, correspondingly. The independence of the strain energy and the elastic dilatation on the DI shape is confirmed. The effect of the aspect ratio and the shape on the elastic fields of the DIs is analyzed. The elastic model of Janus particle and other possible useful applications of solved problems are discussed.

Convergence of the multigrid reduction in time algorithm for the linear elasticity equations

SummaryThis paper presents some recent advances for parallel‐in‐time methods applied to linear elasticity. With recent computer architecture changes leading to stagnant clock speeds, but ever increasing numbers of cores, future speedups will be available through increased concurrency. Thus, sequential algorithms, such as time stepping, will suffer a bottleneck. This paper explores multigrid reduction in time (MGRIT) for an important application area, linear elasticity. Previously, efforts at parallel‐in‐time for elasticity have experienced difficulties, for example, the beating phenomenon. As a result, practical parallel‐in‐time algorithms for this application area currently do not exist. This paper proposes some solutions made possible by MGRIT (e.g., slow temporal coarsening and FCF‐relaxation) and, more importantly, a different formulation of the problem that is more amenable to parallel‐in‐time methods. Using a recently developed convergence theory for MGRIT and Parareal, we show that the changed formulation of the problem avoids the instability issues and allows the reduction of the error using two temporal grids. We then extend our approach to the multilevel case, where we demonstrate how slow temporal coarsening improves convergence. The paper ends with supporting numerical results showing a practical algorithm enjoying speedup benefits over the sequential algorithm.

Analytical solution for cylindrical bending of two-layered corrugated and webcore sandwich panels

We analytically investigate infinitesimal cylindrical bending deformations of two-layered triangular corrugated and webcore linearly elastic sandwich panels by using the mechanics of materials approach and the classical plate theory. The model is validated by comparing its predictions with the solution by the finite element method of the linear elasticity equations for plane strain deformations. The model can accurately capture the secondary bending of the facesheets, manifested as changes in their curvature between the webs and the resulting changes in the axial stresses, from being tensile to possibly compressive, that the commonly-employed homogenization schemes fail to capture. Subsequently, the model is used to analyze several problems with the sandwich panel having a pinned support at the left edge and a roller support at the right edge, and a uniformly distributed load applied on the top facesheet of the panel. It is found that the core plates mostly deform in compression and bending, respectively, for corrugated and web core panels. Furthermore, a significant fraction of the work done by the external load on the structure is absorbed as strain energy of deformations of the core plates near the supports. For four hybrid combinations of corrugated and webcore configurations in two-layered panels, the combination with the upper corrugated core set-up has the least maximum face sheet deflection and axial stress. The analytical technique can be easily extended to multi-layered hybrid configurations and provides quick means of finding efficient strain-energy absorbing hybrid designs.

THE REFINED THEORY OF ELASTIC PLATES AS ASYMPTOTIC APPROACH OF 3D PROBLEM

The refined theory of elastic thin and thick plates is constructed by the asymptotic method for reducing three-dimensional (3D) equations of linear elasticity to two-dimensional ones without the use of any assumptions. The resulting refined theory is much more complicated than the known classical Kirchhoff theory: the required values of the refined theory vary in thickness of the plate by more complex laws; the system of partial differential equations of the refined theory has a higher order than the system of equations of the classical theory. A comparison of the obtained theory with the popular refined theory of Timoshenko and E. Reissner, taking into account the transverse shear deformation is made. It is shown that the inclusion only of the transverse shear deformation is insufficient. In addition to the transverse shear deformation, many additional terms having the same order as the transverse shear deformation must be taken into account.

Diffraction of elastic waves by edges

We investigate the diffraction of singularities of solutions to the linear elastic equation on manifolds with edge singularities. Such manifolds are modeled on the product of a smooth manifold and a cone over a compact fiber. For the fundamental solution, the initial pole generates a pressure wave (p-wave), and a secondary, slower shear wave (s-wave). If the initial pole is appropriately situated near the edge, we show that when a p-wave strikes the edge, the diffracted p-waves and s-waves (i.e. loosely speaking, their singularities together with the singularities of the incoming p-wave are not limits of rays associated to the pressure wave speed which just miss the edge) are weaker in a Sobolev sense than the incident p-wave. We also show an analogous result for an s-wave that hits the edge, and provide results for more general situations.