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This paper deals with a numerical approximation of the elasticity vibration problem based on a potentials decomposition. Decomposing the displacements field into potentials is a well-known tool in elastodynamics that takes advantage of the decoupling of pressure waves and shear waves inside a homogeneous isotropic media. In the spectral problem on a bounded domain, this decomposition decouples the elasticity equations into two Laplacian-like equations that only interact at the boundary. We show that spurious eigenvalues appear when Lagrangian finite elements are used to discretize the problem. Then, we propose an alternative weak formulation which avoids this drawback. A finite element discretization of this weak formulation based again on Lagrangian finite elements is proposed and tested by means of some numerical experiments, which show convergence and absence of spurious modes. [ABSTRACT FROM AUTHOR]

The Note is concerned with a feasibility study of time reversal in a non-homogeneous elastic medium, from data recorded in an acoustic medium. Our aim here is to determine the presence and some physical properties of elastic "inclusions" (unknown, not observable solid objects, characterized by their elastic properties) from partial observations of acoustic waves scattered by these inclusions. A finite element numerical method, based on a variational acousto-elastodynamics formulation, is derived and used to solve the forward, and then, the time-reversed problem. A criterion, derived from the reverse time migration framework, is introduced, to help construct images of the inclusions to be determined. Numerical illustrations on configurations that mimic the breast cancer configuration are proposed, and show that one can differentiate between two inclusions, even with different properties. [ABSTRACT FROM AUTHOR]

Abstract We propose a computational method which improves the identification of sources and scatterers in a two-dimensional elastic medium using the Time Reversal (TR) technique. The identification procedure makes use of partial numerical data available at a small number of sensors. While in realistic applications these data are obtained by lab or field measurements, in the present work they are synthesized by solving the corresponding forward problem. We introduce an "augmentation" procedure that is utilized in addition to the TR technique and investigate its benefits. The augmentation procedure involves the solution of a suitable elliptic problem at selected time steps. In practice, one needs to solve, as a pre-process, a small number of "fundamental" elliptic problems which is equal to the number of measured quantities. A version of the augmentation procedure, which is easier to implement and involves almost no computational cost, is also presented. The local elastic energy in the domain is employed to decide whether the TR solution has properly refocused at the source. Cost functionals, whose minima are sought, are constructed for the source and scatterer identification problems. We conclude that augmentation improves the identification performance, in particular in the presence of noisy measurements. In some cases augmented TR succeeds in the identification whereas non-augmented TR fails. Highlights • The Time Reversal (TR) method for scatterer identification is considered. • An improved TR method is proposed for cases of scarce or very noisy data. • The improvement is achieved by solving an elliptic problem at chosen times. • The associated added computational cost is marginal. • In some cases, the method succeeds in the identification whereas standard TR fails. [ABSTRACT FROM AUTHOR]

Highlights • A new integrated surface unit-cell model at the mesoscopic scale is proposed for 3D four-directional braided composites. • A novel macro-meso finite element modeling strategy for predicting the elastic properties is presented by the periodic boundary conditions. • The key finite element modeling procedures based on the solid unit-cell models are given in detail. Abstract Finite element method based on the mesoscopic scale periodic unit-cell models was often employed to analyze the mechanical properties of 3D braided composites with obvious shell-core structural feature. To obtain the accurate predicted results, the contribution of the interior and surface unit-cell models to the overall mechanical properties should be considered for the finite element modeling. However, how to apply reasonable boundary conditions to the surface unit-cell model is still an interesting issue worth studying, which has great effect on the prediction precision of surface unit-cell model. Meanwhile, it is valuable to develop a novel macro-meso finite element modeling approach combined with the shell-core structural feature of 3D braided composites, which can contribute to improving the prediction precision and saving the computational costs. In this paper, a new integrated surface unit-cell model with the true yarn microstructure at the mesoscopic scale is proposed for 3D four-directional braided composites firstly. Secondly, a novel macro‑meso scale finite element modeling strategy for predicting the elastic properties and the micromechanical responses is presented. Thirdly, the key modeling procedures based on the solid unit-cell models are given. Finally, comparison of the calculated results with the available experimental data validates the effectiveness of the macro‑meso finite element model. The effects of the braiding angle and the fiber volume fraction of specimens on the elastic properties are discussed in detail. Discussion results indicate that the present macro‑meso finite element modeling strategy can be utilized to efficiently predict the mechanical performances of 3D braided composites. [ABSTRACT FROM AUTHOR]

This article presents a general approach akin to domain-decomposition methods to solve a single linear PDE, but where each subdomain of a partitioned domain is associated to a distinct variational formulation coming from a mutually well-posed family of broken variational formulations of the original PDE. It can be exploited to solve challenging problems in a variety of physical scenarios where stability or a particular mode of convergence is desired in a part of the domain. The linear elasticity equations are solved in this work, but the approach can be applied to other equations as well. The broken variational formulations, which are essentially extensions of more standard formulations, are characterized by the presence of mesh-dependent broken test spaces and interface trial variables at the boundaries of the elements of the mesh. This allows necessary information to be naturally transmitted between adjacent subdomains, resulting in coupled variational formulations which are then proved to be globally well-posed. They are solved numerically using the DPG methodology, which is especially crafted to produce stable discretizations of broken formulations. Finally, expected convergence rates are verified in two different and illustrative examples. [ABSTRACT FROM AUTHOR]

In this work we present stabilized finite element methods for the mixed velocity–stress elasticity equations and for its irreducible velocity form. This is done both for the time and frequency domains, the latter being obtained by assuming a harmonic behavior in time. Stabilization methods that belong to the computational framework of the Variational Multi-Scale formulation are used. It is shown that the adequate selection of the algorithmic parameters on which the formulation depends allows one to switch from the primal to the dual functional framework. The performance of the method is tested through several numerical examples, one of which includes a convergence study. [ABSTRACT FROM AUTHOR]

Error estimates for the space-semidiscrete finite element approximation of solutions to initial boundary value problems for linear, second-order hyperbolic systems in bounded polygons [ABSTRACT FROM AUTHOR]

In this article the multiplicative decomposition of the deformation gradient into one part constrained in the direction of the axis of anisotropy and one part describing the directional deformation is proposed. This leads to a clear division of the deformation and stress states in the direction of anisotropy and a remaining part. The decomposition is explained in detail and a constitutive model of hyper-elasticity is proposed for the case of transversal isotropy, where the behavior of the model is investigated with the aid of simple analytical examples. The model is also investigated for inhomogeneous deformation states using high-order finite elements based on hierarchical shape functions showing the sensitivity of the accuracy of the results in the case of anisotropic media. [ABSTRACT FROM AUTHOR]

It is well‐known that the classical Cauchy‐Boltzmann continuum, in spite of its simplicity, has its limitations, e.g. on the simulation of strain localization phenomena or size effects. For a remedy, one may resort to the so‐called generalized continuum theories, see e.g. [2] and references therein. Amongst others, we consider a class of higher order continua, namely the micromorphic continua, where the kinematics are enriched by means of a microstructure undergoing an affine micro deformation. As a drawback, the enriched kinematics lead to additional constitutive equations, increasing the number of material parameters, and therefore making the parameter identification procedure more involved. Here, the parameter identification is viewed as an inverse problem minimizing a least‐square function, see also [3] for a micropolar continuum. For simplicity, we restrict ourselves to a small strain theory. The increasing number of the degrees of freedom and the material parameters clearly motivates an application of adaptive methods. For an application on the direct micromorphic problems, we refer to [5]. Here for the inverse micromorphic problems, we follow the general ansatz by [1] for goal‐oriented error estimation and adaptivity, with which we control the spatial discretization errors of the finite element method. For illustration, numerical examples are presented. [ABSTRACT FROM AUTHOR]

• The data, recorded in the acoustic part of the medium (basically scalar quantities) contain enough information to identify the properties and locations of the inclusions located in the elastic part. This was made possible by introducing criteria allowing us construct images of these inclusions. • Using the same procedure, one can differentiate between benign and malignant inclusions. • This approach is fairly insensitive to noise in the data. The aim of this paper is to study the feasibility of time-reversal methods in a non homogeneous elastic medium, from data recorded in an acoustic medium. We aim to determine, from partial aperture boundary measurements, the presence and some physical properties of elastic unknown "inclusions", i.e. not observable solid objects, located in the elastic medium. We first derive a variational formulation of the acousto-elastic problem, from which one constructs a time-dependent finite element method to solve the forward, and then, the time reversed problem. Several criteria, derived from the reverse time migration framework, are then proposed to construct images of the inclusions, and to determine their locations. The dependence of the approach to several parameters (aperture, number of sources, etc.) is also investigated. In particular, it is shown that one can differentiate between a benign and malignant close inclusions. This technique is fairly insensitive to noise in the data. [ABSTRACT FROM AUTHOR]