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We present a framework for the inverse design of spatially graded phononic materials based on ray tracing. Spatial grading of phononic materials allows the unit cell to vary smoothly in space. Compared to periodic architectures, spatial grading opens up a vast design space that promises new capabilities for manipulating elastic and acoustic waves. However, the use of spatial grading to control wave propagation has been limited to simple gradings, largely due to the bottleneck of modeling efficiency, and to the long-wavelength limit of low frequencies. In this work, ray tracing is exploited as an efficient alternative, around which we develop an optimization framework based on the adjoint state method toward the flexible inverse design of graded phononic materials. We demonstrate the design of graded mass-spring networks for objectives including focusing all directions of a point source, broadband focusing of a plane wave, and frequency sorting. These objectives are out of reach of periodic phononic materials, highlighting the high potential of spatially graded phononic materials. Moreover, our results demonstrate the importance of wave dispersion, which is classically neglected in the long-wavelength limit of elastodynamics. [ABSTRACT FROM AUTHOR]

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [ABSTRACT FROM AUTHOR]

The optimal local truncation error method (OLTEM) with unfitted Cartesian meshes recently developed for the scalar wave and heat equations for heterogeneous materials is extended to a more complex case of a system of the elastodynamics PDEs. Compact 9-point stencils (similar to those for linear finite elements) are used for OLTEM. Compared to our previous results, a new approach is used for the calculation of the right-hand side of the stencil equations due to body forces. It significantly simplifies the analytical derivations of OLTEM for time-dependent problems. There are no unknowns on interfaces between different materials; the structure of the global semi-discrete equations for OLTEM is the same for homogeneous and heterogeneous materials. For the first time we have also developed OLTEM with the diagonal mass matrix. In contrast to many known approaches with some ad-hoc calculations of the diagonal mass matrix, OLTEM offers a rigorous approach which is a particular case of OLTEM with the non-diagonal mass matrix. Another novelty of the article is a new post-processing procedure for the accurate calculations of stresses. It includes the same compact 9-point stencils as those in basic computations and uses the accelerations and the displacements at the grid points along with the PDEs for the stress calculations. OLTEM yields accurate numerical results for heterogeneous materials with big contrasts in the material properties of different components. Numerical experiments for elastic heterogeneous materials show: a) at the same number of degrees of freedom (dof), OLTEM with unfitted Cartesian meshes is more accurate than linear finite elements with similar stencils and conformed meshes; at the engineering accuracy of 0.1 % for the displacements, OLTEM reduces the number of dof by more than 20 times; at the engineering accuracy of 0.1 % for the stresses, OLTEM with the new post-processing procedure reduces the number of dof by more than 104 times compared to linear finite elements; b) at the same number of dof, OLTEM with unfitted Cartesian meshes is even more computationally efficient than high-order finite elements with much wider stencils and conformed meshes. This will lead to a huge reduction in the computation time for elastodynamics problems solved by OLTEM and will allow the direct computations of some complex wave propagation and structural dynamics problems for heterogeneous materials without the scale separation. [ABSTRACT FROM AUTHOR]

Equations of motion are derived for (visco)elastic, self-gravitating and variably rotating planets. The equations are written using a decomposition of the elastic motion that separates the body's elastic deformation from its net translational and rotational motion as far as possible. This separation is achieved by introducing degrees of freedom that represent the body's rigid motions; it is made precise by imposing constraints that are physically motivated and that should be practically useful. In essence, a Tisserand frame is introduced exactly into the equations of solid mechanics. The necessary concepts are first introduced in the context of a solid body, motivated by symmetries and conservation laws, and the corresponding equations of motion are derived. Next, it is shown how those ideas and equations of motion can readily be extended to describe a layered fluid–solid body. A possibly new conservation law concerning inviscid fluids is then stated. The equilibria and linearization of the fluid–solid equations of motion are discussed thereafter, along with new equations for use within normal-mode coupling calculations and other Galerkin methods. Finally, the extension of these ideas to the description of multiple, interacting fluid–solid planets is qualitatively discussed. [ABSTRACT FROM AUTHOR]

In this article we present a boundary element formulation for time dependent elastodynamic wave propagation in 2D interior domains with inhomogeneous boundary conditions. These are imposed in a weak sense, allowing the unified implementation of a wide range of boundary data, from pure Dirichlet to pure Neumann and mixed ones. Numerical experiments indicate that the considered approach is effective and reliable. We discuss several numerical results which, compared to those obtained by the strong imposition of the boundary data, highlight the equivalence in the behavior of the error. [ABSTRACT FROM AUTHOR]

This work focuses on the numerical performance of HHT- α and TR-BDF2 schemes for dynamic frictionless unilateral contact problems between an elastic body and a rigid obstacle. Nitsche's method, the penalty method, and the augmented Lagrangian method are considered to handle unilateral contact conditions. Analysis of the convergence of an opposed value of the parameter α ~ for the HHT- α method is achieved. The mass redistribution method has also been tested and compared with the standard mass matrix. Numerical results for 1D and 3D benchmarks show the functionality of the combinations of schemes and methods used. [ABSTRACT FROM AUTHOR]

The thermodynamic model of viscoelastic deformable solids at finite strains is formulated in a fully Eulerian way in rates. Also, effects of thermal expansion or buoyancy due to evolving mass density in a gravity field are covered. The Kelvin–Voigt rheology with a higher-order viscosity (exploiting the concept of multipolar materials) is used, allowing for physically relevant frame-indifferent stored energies and for local invertibility of deformation. The model complies with energy conservation and Clausius–Duhem entropy inequality. Existence and a certain regularity of weak solutions are proved by a Faedo–Galerkin semi-discretization and a suitable regularization. Subtle physical limitations of the model are illustrated on thermally expanding neo-Hookean materials or materials with phase transitions. [ABSTRACT FROM AUTHOR]

Metamaterials (MMs), which include phononic crystals (PCs) as a particular type, exhibit anomalous wave propagation properties through artificial design of topologies or lattice forms of unit‐cells. Recent advancements in MMs signify an ascendant research trend, providing promising design ideas and means for unprecedented wave propagation properties. The imperative for on‐demand, real‐time active control of wave propagation underscores the significance of tunable manipulation of acoustic/elastic waves, promoting the design and development of tunable MMs. Furthermore, the versatility of intelligent materials and their ongoing development and innovation contribute significantly to the emergence of diverse intelligent MMs. This comprehensive survey provides an overview of recent advancements and current research trends in the interdisciplinary field of intelligent MMs with electro‐/magneto‐mechanical couplings. The primary objective of the review is to emphasize significant progress in agile manipulation of acoustic/elastic waves in electro‐/magneto‐mechanical coupled MMs, followed by an in‐depth exploration of intelligent metasurfaces, topological MMs, non‐Hermitian parity‐time symmetric wave systems, odd elastic MMs, and spatiotemporally modulated MMs. Special emphasis is given to multi‐field coupling effects. The review concludes with a summary and outlines potential prospects, offering a timely and informative guide for future studies on actively tunable PCs and MMs in practical engineering applications. [ABSTRACT FROM AUTHOR]

Founded on a novel analytical formulation that led to a rigorous yet compact path-integral representation of the time-domain elastodynamic half-space Green's function, a unified analysis of the possible occurrence of different singular wavefront behaviour in the response under arbitrary impulsive internal or surface point loads at arbitrary source-receiver locations is presented. With the decomposition of the general solution into distinct initiating and reflected wave group integrals that share a common factored format and simple contour definitions, the mathematical framework is shown to allow a straightforward identification of the specific conditions and the particular wave groups that are responsible for the singular wavefront phenomena without resorting to advanced analytic function theories or asymptotic methods. Analytic characterizations of the nature, strength and direction of all intrinsic singular wavefront behaviours of the three-dimensional Green's function in three canonical cases of source-receiver configurations are given in a dual integral-closed form format to facilitate their theoretical understanding as well as computational applications. Graphical illustrations of their variation with the source-receiver configuration and the medium's Poisson's ratio together with relevant comparison and clarifications of some classical treatments are included. [ABSTRACT FROM AUTHOR]

The thermodynamic model of viscoelastic deformable magnetic materials at finite strains is formulated in a fully Eulerian way in rates. The Landau theory applied to a ferro-to-paramagnetic phase transition, the gradient theory (due to an exchange energy) for magnetization with general mechanically dependent coefficient, hysteresis in magnetization evolution by the Gilbert equation involving an objective corotational time derivative of magnetization, and the demagnetizing field are considered in the model. The Kelvin–Voigt viscoelastic rheology with a higher-order viscosity (exploiting the concept of multipolar materials) is used, allowing for physically relevant frame-indifferent stored energies and for local invertibility of deformation. The model complies with energy conservation and Clausius–Duhem entropy inequality. An existence and a certain regularity of weak solutions are proved by a Faedo–Galerkin semi-discretization and a suitable regularization. [ABSTRACT FROM AUTHOR]

Recently, Murthy et al. (Commun Comput Phys 2:23, 2017. http://dx.doi.org/10.4208/cicp.OA-2016-0259) and Escande et al. (Lattice Boltzmann method for wave propagation in elastic solids with a regular lattice: theoretical analysis and validation, 2020. arXiv.doi:1048550/ARXIV.2009.06404. arXiv:2009.06404) adopted the Lattice Boltzmann Method (LBM) to model the linear elastodynamic behaviour of isotropic solids. The LBM is attractive as an elastodynamic solver because it can be parallelised readily and lends itself to finely discretised simulations of dynamic effects in continua, allowing transient phenomena such as wave propagation to be modeled efficiently. This work proposes simple local boundary rules which approximate the behaviour of Dirichlet and Neumann boundary conditions with an LBM for elastic solids. The boundary rules are shown to be consistent with the target boundary values in the first order. An empirical convergence study is performed for the transient tension loading of a rectangular plate, with a Finite Element (FE) simulation being used as a reference. Additionally, we compare results produced by the LBM for the sudden loading of a stationary crack with an analytical solution from Freund (Dynamic fracture mechanics. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511546761). [ABSTRACT FROM AUTHOR]

Engineered piezoelectric metamaterials can possess electro-momentum coupling between the macroscopic momentum and electric stimuli. This indicates the applicability of such metamaterials for wave scattering with an extra design degree of freedom, in the same way as Willis materials. To fully utilize this novel bianisotropy, we derive for the first-time theoretical bounds on electro-momentum coupling in wave scattering via energy conservation. As this coupling acts on both elastodynamics and electromagnetics, the polarizability tensor is generalized to fill their link in the bounds. Our derived bounds are verified via analytical scattering solutions. Results show that the bianisotropic scattering can be of the same order as the non-bianisotropic terms via the aid of electro-momentum coupling even for small Willis coupling. We further reveal the possibility of using electro-momentum coupling for tunable cloaking. This sheds light on the promising potential of piezoelectric metamaterials for tunable scattering devices whose bianisotropy can be modulated by external electric stimuli. [ABSTRACT FROM AUTHOR]

This article focuses on computational studies evaluating the influence of crystallinity, residual stresses, and out-of-plane (OOP) deterministic switching on Terfenol-D nano/microstructures. The computational models use both coupled and uncoupled Landau–Liftshitz–Gilbert equations with elastodynamics to study strain-induced magnetization reorientation. A Voronoi tessellation approach models the crystal distribution in the microstructures subjected to residual stresses with good agreement to experimental data including large changes in coercivity values, i.e., from 100 to 3000 Oe. Parametric studies show how the coercivity is manipulated with residual stresses, including a magnetoelastically induced perpendicular-magnetic-anisotropy (PMA), important for memory applications. Additional parametric studies focus on epitaxially deposited micro-disks, revealing that residual stresses can create magnetoelastically dominant easy axes along the ⟨ 110 ⟩ directions, which are energetically favorable relative to the intrinsic ⟨ 111 ⟩ magnetocrystalline easy axes. Modification of the global easy axis is used to design a strain-mediated multiferroic composite consisting of a 20 nm epitaxially deposited Terfenol-D memory bit with PMA grown on a PZT substrate. The multiferroic disk achieves OOP deterministic clocking with an applied voltage. [ABSTRACT FROM AUTHOR]

This manuscript presents an efficient solver for the linear system that arises from the hierarchical Poincar\'e--Steklov (HPS) discretization of three dimensional variable coefficient Helmholtz problems. Previous work on the HPS method has tied it with a direct solver. This work is the first efficient iterative solver for the linear system that results from the HPS discretization. The solution technique utilizes GMRES coupled with a locally homogenized block-Jacobi preconditioner. The local nature of the discretization and preconditioner naturally yield the matrix-free application of the linear system. Numerical results illustrate the performance of the solution technique. This includes an experiment where a problem approximately 100 wavelengths in each direction that requires more than a billion unknowns to achieve approximately 4 digits of accuracy takes less than 20 minutes to solve. [ABSTRACT FROM AUTHOR]

To evaluate the mechanical behavior and the load transfer mechanisms of a geosynthetic-reinforced pile-supported (GRPS) embankment under traffic load, an analytical model is developed based on elastodynamic theory, and its solution is derived rigorously. Since the present model is continuum based, the displacement and stress distributions in the embankment can be obtained analytically. Additionally, the present model takes the anisotropic property of the reinforced layer into consideration. The governing equations of the model are solved by introducing Fourier expansions of the field variables and making use of the theory of differential equations. With the boundary and interface conditions, the solution of the system is determined. The influences of some critical parameters are investigated. Here we show the load velocity has a significant influence on the response of the system, especially when it approaches the critical point. The present results suggest that the maximum efficiency of a single pile Max{EPi} at v = 100 m/s is almost 2 times higher than its value at v = 1 m/s. Some other parameters, such as pile cap spacing, embankment height, and the stiffness ratio between subsoil and piles, also affect the distributions of EPi , while the total pile efficiency EP only depends on the supporting properties. Some practical suggestions are also presented based on the parametric studies. [ABSTRACT FROM AUTHOR]

In our previous study (Part I), the anti-plane steady-state hyperbolic mode III fracture of a magneto-flexoelectric material was solved for the displacement, the polarization, and the magnetic fields. The solution, however, was based on the assumption of the development of strain discontinuities, and the propagation of the crack-tip was related to a critical shear strain. However, in the current study, the asymptotic details of the fields close to the crack-tip were investigated. The asymptotic analysis assumes strain continuity at the crack-tip (discontinuity in the strain gradients) and reveals the existence of a positive dynamic J-integral. The asymptotic analysis was performed not only for hyperbolic but also for elliptic conditions, and the energy release rate was calculated as a function of the crack-tip velocity in both regimes. These results are very different from those predicted by classical singular elastodynamics, where the dynamic J-integral is zero when super-shear is attained and there can be only an elliptic solution. Moreover, the results are very useful for couple-stress elastodynamics where equivalent length scales are present due to the analogy with flexoelectricity. [ABSTRACT FROM AUTHOR]

We consider the retarded potential boundary integral equation, arising from the 3D elastic (vector) wave equation problem, endowed with a Dirichlet condition on the boundary and null initial conditions. For its numerical solution, we employ a weak formulation related to the energy of the system and we discretize it by a Galerkin-type boundary element method (BEM). This approach, called energetic BEM, has been already applied in the context of time-domain acoustic (scalar) wave propagation and it has revealed accurate and stable even on large time intervals of analysis. In particular, when standard (constant) shape functions for time discretization are employed, the double integration in time can be performed analytically. Then, one is left with the task of evaluating double space integrals, whose integration domains are generally delimited by the wave fronts of the primary and the secondary waves. Since the accurate computation of the integrals involved in the numerical scheme is a key issue for the stability of the method, we propose an efficient evaluation strategy, based on the exact detection of the integration domain. The presented numerical tests show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In recent years, the development of lattice Boltzmann methods (LBMs) for solids has gained attention. Fracture mechanics as a viable application for these methods has been presented before, albeit for mode III cracks in the context of an LBM for antiplane shear deformation. The performance of the LBM itself is promising, while the usage of a regular lattice simplifies the modelling of fractures significantly. Recent advancements in LBMs for solids, especially the description of Dirichlet‐ and Neumann‐type boundary conditions, now make it possible to extend the LBM simulation of crack propagation to the plane strain case with modes I and II crack opening, including growth with non‐uniform speed in arbitrary directions. For this, the configurational force acting on a crack tip is utilised. The definition of the moments of the LBM, which are based on the balance laws of continuum mechanics, render the evaluation of macroscopic fields in the configuration straightforward. In this work, the general in‐plane case of dynamic crack propagation is shown and necessary considerations for the implementation are discussed. Lastly, numerical examples showcase the capabilities of the proposed method to model dynamic fractures and establish a proof‐of‐concept. [ABSTRACT FROM AUTHOR]

We provide a fully nonlinear port‐Hamiltonian formulation for discrete elastodynamical systems as well as a structure‐preserving time discretization. The governing equations are obtained in a variational manner and represent index‐1 differential algebraic equations. Performing an index reduction, one obtains the port‐Hamiltonian state space model, which features the nonlinear strains as an independent state next to position and velocity. Moreover, hyperelastic material behavior is captured in terms of a nonlinear stored energy function. The model exhibits passivity and losslessness and has an underlying symmetry yielding the conservation of angular momentum. We perform temporal discretization using the midpoint discrete gradient, such that the beneficial properties are inherited by the developed time stepping scheme in a discrete sense. The numerical results obtained in a representative example are demonstrated to validate the findings. [ABSTRACT FROM AUTHOR]

In order to make good use of the inherent advantages of the FEM and the BEM formulations, an iterative decomposition coupling algorithm between time-domain FEM and time-domain BEM with independent spatial discretization on the interface is proposed for elastodynamic problems. In the coupling process, based on the conditions of the displacement continuity and the force equilibrium on the interface, the force and the displacement converting matrices are obtained. Data are exchanged on the interface between the time-domain FEM variables and the time-domain BEM variables through the converting matrices, instead of the interpolation calculation for the variables between the two sub-domains. Releasing the limitation of the interpolation function of variables on the interface, the proposed coupling algorithm is easily manipulated in data exchange between nodal variables on the interface. Additionally, the independent spatial discretization on the interface suitable for one's own boundary conditions and material properties in both the TD-FEM and TD-BEM sub-domains, the proposed coupling algorithm is of good versatility and computational efficiency. Three benchmark examples are presented to verify the correctness and efficiency of the proposed coupling algorithm. [ABSTRACT FROM AUTHOR]

In recent years, there has been increasing attention on the development of non-destructive evaluation (NDE) methods using guided waves for long-length materials such as thin plates and pipes. The guided waves are capable of long-distance propagation in thin plates and pipes, and they exhibit properties such as multimodality and dispersion. These characteristics of the guided waves make inspection using guided waves challenging. In this study, we apply a 2-D elastodynamic time-reversal method to detect surface breaking cracks of a thin plate where guided waves are present. The finite element method (FEM) is used to calculate the scattered waves from surface breaking cracks and their corresponding time-reversal waves. We also employ topological sensitivity as an assessment index for detecting surface breaking cracks using the time-reversal method. As numerical examples, we demonstrate guided wave propagation, scattering, and the time-reversal wave propagation obtained by using the FEM. Finally, we present the results of surface breaking crack detection in a thin plate and discuss the validity and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Accurate modeling of elastic wavefields in 3-D anisotropic media is important for many seismic processing and inversion applications. However, efficient wavefield simulation for tilted transversely isotropic (TTI) media and, especially, for orthorhombic and lower symmetries remains challenging. Finite-difference (FD) implementations using centered Taylor-series coefficients on singly staggered grids suffer from reduced numerical accuracy due to problems in computing the partial wavefield derivatives in TTI or tilted orthorhombic (TOR) media, as well as in enforcing the free-surface (zero-traction) boundary conditions. To address these issues, we develop a 3-D mimetic FD (MFD) algorithm for arbitrarily anisotropic media that uses a fully-staggered-grid strategy. This CUDA-based algorithm is implemented on graphics processing units (GPUs) to leverage the massive parallelism of this computer architecture. For multi-GPU parallelization, we employ the CUDA-aware message passing interface (MPI) library to exploit the remote direct memory access (RDMA) feature for buffer transfers. Weak- and strong-scaling tests on up to eight DGX NVIDIA A100 nodes (64 GPUs in total) demonstrate that the proposed multi-GPU implementation achieves a quasi-linear computational speedup with over 98% efficiency for large industrial-scale models of size in excess of 1.7 × 10 10 grid points. [ABSTRACT FROM AUTHOR]

In this paper, we present an immersed finite element (IFE) method for solving the elastodynamics interface problems on interface-unfitted meshes. For spatial discretization, we use vector-valued P 1 and Q 1 IFE spaces. We establish some important properties of these IFE spaces, such as inverse inequalities, which will be crucial in the error analysis. For temporal discretization, both the semi-discrete and the fully discrete schemes are derived. The proposed schemes are proved to be unconditionally stable and enjoy optimal rates of convergence in the energy, L 2 and semi- H 1 norms. Numerical examples are designed to verify our theoretical analysis and to demonstrate the stability and robustness of our schemes. [ABSTRACT FROM AUTHOR]

We study the vanishing shear viscosity limit and incompressible limit of two dimensional compressible dissipative elastodynamics systems near the equilibrium. The large value of the volume viscosity forces the limit system to be incompressible, and the vanishing shear viscosity indicates that the limit system is inviscid. Due to the weak dispersive estimate in dimensions two and the lack of the null structures in nonlinear terms, the generalised energy method (Klainerman and Sideris in Commun Pure Appl Math 49(3):615–666, 1996; Sideris in Ann Math 151(2):849–874, 2000; Sideris and Thomases in Commun Pure Appl Math 58(6):750–788, 2005), combined with the "ghost weight" method in (Alinhac in Invent Math 145(3):597–618, 2001), guarantees the global-in-time convergence from compressible dissipative elastodynamics to incompressible elastodynamics with the help of vector fields. [ABSTRACT FROM AUTHOR]

A hybrid implicit-explicit (IMEX) discontinuous Galerkin spectral element time domain (DG-SETD) algorithm is proposed to simulate 3D elastic wave propagation in inhomogeneous media. In this method, the original problem can be divided into a number of well designed subdomains, and the mesh generation of different subdomains is completely independent, thus allowing flexible spatial discretization of complex geometry. The neighboring subdomains are connected by a Riemann transmission condition (RTC), and spectral elements with different interpolation orders can be used in different subdomains to maximize the computational efficiency of multiscale problems to facilitate parallel computing for different subdomains. In particular, the explicit or implicit time iteration scheme can be appropriately selected for a subdomain according to the size of its mesh elements to increase the global time step increment, thus giving higher computational efficiency: For subdomains with coarse meshes, the explicit time integration scheme is used and the time step increment is limited by the Courant−Friedrichs−Lewy (CFL) stability condition; for subdomain with fine structures and thus fine meshes, an implicit time integration scheme is used so that a large time step increment can be used without affecting the stability. In addition, a jump condition of displacement and velocity is introduced to accurately simulating fractures and faults, including lossless and viscous fractures with plane, curve or cross structures. This avoids the volume modeling of the extremely thin fracture structures, and effectively reduces the number of degrees of freedoms (DoFs) in the discretized system without the loss of accuracy. The accuracy, robustness and efficiency of the DG-SETD algorithm are demonstrated by quantitative comparisons of the waveforms with the commercial finite element software COMSOL. [ABSTRACT FROM AUTHOR]

This paper concerns the time growth of the highest-order energy of the systems of incompressible isotropic Hookean elastodynamics in two space dimensions. The global well-posedness of smooth solutions near equilibrium was proved by Lei [Comm. Pure Appl. Math., 69 (2016), pp. 2072-2106] where the highest-order generalized energy may have certain growth in time. We develop novel energy estimate strategies based on the inherent structures and show that the highest-order generalized energy is uniformly bounded for all the time. [ABSTRACT FROM AUTHOR]

This paper presents a concurrent topology optimization of multi-scale composite structures subjected to general time-dependent loads for minimizing dynamic compliance. A three-field density-based method is adopted to implement the concurrent topological design, with macroscopic effective properties of the microstructure evaluated through energy-based homogenization method (EBHM). Transient response is obtained from the two-scale finite element analysis with the HHT-α approach as an implicit time integration procedure. Design sensitivities are formulated employing the adjoint variable method (AVM) based on two main philosophies: "discretize-then-differentiate" and "differentiate-then-discretize" approaches, respectively. The method of moving asymptotes is adopted to update the design variables at two scales. Several benchmark examples are presented to demonstrate that the "discretize-then-differentiate" AVM attains consistent sensitivities in an inherent manner such that the resulting optimal topology is more efficient when compared with the "differentiate-then-discretize" AVM. Moreover, the potential of the proposed method for concurrent dynamic topology optimization problems under general time-dependent loads is also highlighted. [ABSTRACT FROM AUTHOR]

We show that fine phase mixtures arise in the initial-boundary value problem for a class of equations of non-convex elastodynamics in one space dimension. Specifically, we prove that there are infinitely many local-in-time Lipschitz weak solutions to such a problem that exhibit immediate fine-scale oscillations of the strain whenever the range of the initial strain has a nonempty intersection with the elliptic regime. Consequently, such solutions are nowhere C 1 in the part of the space-time domain with fine phase mixtures, but are smooth in the other part of the domain. [ABSTRACT FROM AUTHOR]

• The first analytical poroelastodynamics solution for an isotropic fluid-saturated porous cylinder. • Systematic interpretation of the mechanisms for the elastic moduli dispersion. • Comprehensively investigating effects of frequency, sample size, poromechanical parameters, and boundary conditions. • Excellent matches between the analytical solution and laboratory measurements of a sediment rock and a limestone. There are extensive laboratory measurements of elastic moduli dispersion of porous materials, yet the mechanisms for the dispersion are not fully understood. Related analytical solutions are missing in the literature, potentially due to complex mathematical derivations. In this paper, we use Biot's theory of poroelastodynamics (PED) and present the first analytical solution for an isotropic fluid-saturated porous cylinder subject to a forced deformation test. This is done by introducing Helmholtz decomposition, two scalar potentials, and two vector potentials to decouple the original governing partial differential equations. The methods of matrix diagonalization and separation of variables are then used to solve the decoupled equations. The elastodynamics (ED) solution is also presented. We demonstrate the coupled responses of displacement, pore pressure, and stress and interpret systematically the mechanisms for the elastic moduli dispersion. Furthermore, we investigate comprehensively the effects of loading frequency, material's poromechanical characteristics, sample size, and boundary conditions on the elastic moduli dispersion. Finally, excellent matches are found between the analytical solution and published measurements of the dynamic Young's moduli and Poisson's ratios of a water-saturated clastic sediment rock and a glycerin-saturated limestone. The analytical solution captures the materials' elastic moduli dispersion at both seismic and sonic frequencies. This work should be of interest to scientists and engineers who work on dynamics, vibration, and poromechanics. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of interaction of waves originating from the Riemann problem centered at two different points for a system of equations modeling propagation of elastic waves. The system consists of two equations for (u,σ)$$ \left(u,\sigma \right) $$, where u$$ u $$ is the velocity and σ$$ \sigma $$ is the stress and is strictly hyperbolic and nonconservative. The study of interaction of waves is one of the most important steps in the construction of a global solution with initial data in the space of functions of bounded variation using an approximation procedure like Glimm's scheme. This amounts to constructing a solution with initial data consisting of three states (uL,σL)$$ \left({u}_L,{\sigma}_L\right) $$, (um,σm)$$ \left({u}_m,{\sigma}_m\right) $$, and (uR,σR)$$ \left({u}_R,{\sigma}_R\right) $$. Usually, this analysis is done for the states which are in a small neighborhood of a fixed state. Here, we get an explicit formula for the solution of the system when the data lie in the level sets of Riemann invariants. The speciality of the work is that we do not assume smallness conditions on the initial data. [ABSTRACT FROM AUTHOR]

Three-dimensional dynamic problem for a layered orthotropic elastic shell with free upper face is considered. The interfaces between the layers are assumed to be in perfect contact and the displacements of one of the interfaces are prescribed. A long-wave asymptotic solution is constructed and the thickness resonances are determined. The obtained results can be applied in the evaluation of some parameters of the earthquakes. [ABSTRACT FROM AUTHOR]

A model of saturated hyperelastic porous solids at large strains is formulated and analyzed. The material response is assumed to be of a viscoelastic Kelvin--Voigt type, and inertial effects are considered, too. The flow of the diffusant is driven by the gradient of the chemical potential and is coupled to the mechanics via the occurrence of swelling and squeezing. Buoyancy effects due to the evolving mass density in a gravity field are covered. Higher-order viscosity is also included, allowing for physically relevant stored energies and local invertibility of the deformation. The whole system is formulated in a fully Eulerian form in terms of rates. The energetics of the model is discussed, and the existence and regularity of weak solutions is proved by a combined regularization-Galerkin approximation argument. [ABSTRACT FROM AUTHOR]

We present a new convolutional neural network (CNN)-based element-wise classification method to detect a random number of voids with arbitrary shapes in a two-dimensional (2D) plane-strain solid subjected to elastodynamics. We consider that an elastic wave source excites the solid including a random number of voids, and wave responses are measured by sensors placed around the solid. We present a CNN for resolving the inverse problem, which is formulated as an element-wise classification problem. The CNN is trained to classify each element into a regular or void element from measured wave signals. Element-wise binary classification enables the identification of targeted voids of any shapes and any number without prior knowledge or hint about their locations, shape types, and numbers, while existing methods rely on such prior information. To this end, we generate training data consisting of input-layer features (i.e., measured wave signals at sensors) and output-layer features (i.e., element types of all elements). When the training data are generated, we utilize the level-set method to avoid an expensive remeshing process, which is otherwise needed for each different configuration of voids. We also analyze how effectively the CNN performs on blind test data from a non-level-set wave solver that explicitly models the boundary of voids using an unstructured, fine mesh. Numerical results show that the suggested approach can detect the locations, shapes, and sizes of multiple elliptical and circular voids in the 2D solid domain in the test data set as well as a blind test data set. [ABSTRACT FROM AUTHOR]

The low Mach number limit of the nonisentropic compressible Hookean elastodynamic equations is rigorously proved with respect to well‐prepared initial data. We introduce certain suitable seminorms to obtain the uniform estimate of solutions, for which the critical point is to cancel the higher order derivate terms caused by the coupling of velocity and deformation gradient. [ABSTRACT FROM AUTHOR]

Based on the one-dimensional elastic theory, the radial vibrations of cylindrical isotropic embedded in an elastic medium are studied in the paper. The effect of the magnetic field of the radial vibrations of an elastic hollow cylinder with rotation is researched as well. The one-dimensional equation of elastodynamic is solved in terms of radial displacement. The frequency equation is obtained when the boundaries are free and fixed; the mixed boundary condition is numerically examined. The determination is concerned with the eigenvalues of the natural frequency of the radial vibrations in the case of harmonic vibrations. The effect of the magnetic field and rotation on the natural frequencies were explored. It was shown that the dispersion curves of guided waves were significantly influenced by the magnetic field and rotation of the elastic cylinder. Numerical results are given and graphically illustrated in each considered case. The natural frequencies and mode shapes are calculated numerically, and the effects of rotating, magnetic field, and variable thickness are discussed. It is observed that an increase of the magnetic parameter, as well as the rotation parameters, brings results closer to the classical cylinder theory results. Furthermore, the current study can be applied to the design of microplates and nanoplates and their optimal usage. The results indicate that the effects of the magnetic field and rotation are very pronounced. [ABSTRACT FROM AUTHOR]

In the present contribution, a novel mixed formulation aimed at the energy‐momentum consistent simulation of coupled nonlinear electro‐thermo‐elastodynamic systems, in particular dielectric elastomer actuators, is proposed. It is essentially based on a mixed framework for elastodynamics in the case of polyconvex stored energy functions. In accordance with this framework, the properties of the rediscovered tensor cross product are exploited in a first step to derive a mixed formulation via a Hu‐Washizu type extension of the stored energy function. Afterwards, the corresponding strong form is derived and supplemented with the (initial) boundary value problems of thermodynamics and electrostatics. By additionally choosing an appropriate polyconvexity‐inspired energy density function, this procedure leads to a fully coupled electro‐thermo‐elastodynamic formulation that benefits from the properties of the underlying mixed framework. Furthermore, an energy‐momentum consistent time integration scheme is proposed for the novel framework, where discrete derivatives in the sense of Gonzalez are employed. The formulation is second‐order accurate and stable even for large time step sizes. Eventually, the performance of the novel formulation is illustrated in a numerical example. [ABSTRACT FROM AUTHOR]

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]

Multi-axial perfectly matched layer (M-PML), known to have lost the perfect-matching property owing to multi-axial coordinate stretching, has been numerically validated to be long-time stable and it is thus used extensively in linear anisotropic wave simulation and in isotropic cases where the PML becomes unstable. We are concerned with the construction of the M-PML for anisotropic wave simulation based on a second order wave equation implemented with the displacement-based numerical method. We address the benefit of the incorrect chain rule, which is implicitly adopted in the previous derivation of the M-PML. We show that using the frequency-shifted stretching function improves the absorbing efficiency of the M-PML for near-grazing incident waves. Then, through multi-axial complex-coordinate stretching the second order anisotropic wave equation in a weak form, we derive a time-domain multi-axial unsplit frequency-shifted PML (M-UFSPML) using the frequency-shifted stretching function and the incorrect chain rule. A new approach is provided to reduce the number of memory variables needed for computing convolution terms in the M-UFSPML. The obtained M-UFSPML is well suited for implementation with a finite element or the spectral element method. By providing several typical examples, we numerically verify the accuracy and long-time stability of the implementation of our M-UFSPML by utilizing the Legendre spectral element method. [ABSTRACT FROM AUTHOR]

We study the propagation of shear waves in an anisotropic incompressible medium composed of an elastic fiber-reinforced material. We first derive the general determining equations of such motions when the fibers are arbitrarily arranged with respect to the direction of propagation of the waves. Then, we examine the "standard reinforced model" so as to root the nonlinearity of the dynamic problem directly to the presence of the reinforcing fibers, and finally, we consider an asymptotic model in terms of the limit of finite but small amplitude of the waves. Clearly, if the fibers are arranged along the direction of propagation of the waves, the mechanical behavior of the material will be isotropic. When the fibers are tilted an amount of the same order of the amplitude of the wave with respect to this direction, the asymptotic first-order reduced system is characterized by a peculiar mixed second-/third-order nonlinearity. This contrasts to what occurs for larger tilting angles where only quadratic nonlinearities must be considered. In the latter case, as we may expect, in the asymptotic limit, we reduce the system to an inviscid classical Burger's equation. Our results are fundamental to the study of the dynamics of fiber-reinforced materials and equally when we consider dissipative and/or dispersive effects. [ABSTRACT FROM AUTHOR]

We investigate topology optimization of the elastodynamic structures under impact loading with transient stress constraints. In contrast to existing methods that apply a time-dependent prescribed traction force, we model the impact problem by directly imposing a prescribed loading rate on the boundary over time. It allows us to carry out topology optimization for impacted structures with strongly acyclic and transient characteristics. We minimize the volume fraction of the structure while constraining the maximum transient von Mises stress. An additional static compliance constraint is introduced to guarantee the validity of the transient analysis-based design. The sensitivity of maximum transient von Mises stress is then derived and validated, in which the discretize-then-differentiate method is employed for the dynamic problem. A series of benchmark problems are investigated, by which the effectiveness of the proposed method is illustrated. The evolution of the maximum stress shows the significance of highlighting transient behavior in the topology optimization of impacted structures. [ABSTRACT FROM AUTHOR]

We are concerned with the vortex sheet solutions for the three-dimensional compressible isentropic elastic flows. This is a nonlinear hyperbolic problem with a characteristic free boundary. Compared with the analysis in two dimensions, this added dimension leads to more complicated frequency interactions between the effects of elasticity and the fluid velocity, making the stability analysis more challenging. Through a very delicate examination of the Lopatinskiĭ determinant of the linearized boundary value problem, necessary and sufficient conditions are established for the linear stability of the planar vortex sheet solutions. These conditions are closely related to the geometric properties of the elastic deformation gradient and provide the first stability criterion justifying the stabilization effect of elasticity on the compressible vortex sheets in the three-dimensional elastodynamics. In contrast to the two-dimensional isentropic elastic fluids, we find that the stability can only hold in the subsonic region for the three-dimensional vortex sheets. [ABSTRACT FROM AUTHOR]

In this work, we present a high-order Discontinuous Galerkin time integration scheme for second-order (in time) differential systems that typically arise from the space discretization of the elastodynamics equation. By rewriting the original equation as a system of first-order differential equations we introduce the method and show that the resulting discrete formulation is well-posed, stable, and retains a super-optimal rate of convergence with respect to the discretization parameters, namely the time step and the polynomial approximation degree. A set of two- and three-dimensional numerical experiments confirm the theoretical bounds. Finally, the method is applied to real geophysical simulations. [ABSTRACT FROM AUTHOR]

This paper tracks the development of lattice models that aim to describe linear elasticity of solids and the field equations of which converge asymptotically toward those of isotropic continua, thus showing the connection between discrete and continuum. In 1759, Lagrange used lattice strings/rod dynamics to show the link between the mixed differential-difference equation of a one-dimensional (1D) lattice and the partial differential equation of the associated continuum. A consistent three-dimensional (3D) generalization of this model was given much later: Poincaré and Voigt reconciled the molecular and the continuum approaches at the end of the nineteenth century, but only in 1912 Born and von Kármán presented the mixed differential-difference equations of discrete isotropic elasticity. Their model is a 3D generalization of Lagrange's 1D lattice and considers longitudinal, diagonal and shear elastic springs among particles, so the associated continuum is characterized by three elastic constants. Born and von Kármán proved that the lattice equations converge to Navier's partial differential ones asymptotically, thus being a formulation of continuous elasticity in terms of spatial finite differences, as for Lagrange's 1D lattice. Neglecting shear springs in Born–Kármán's lattice equals to Navier's assumption of pure central forces among molecules: in the limit, the lattice behaves as a one-parameter isotropic solid ("rari-constant" theory: equal Lamé parameters, or, equivalently, Poisson's ratio υ = 1 / 4 ). Hrennikoff and McHenry revisited the lattice approach with pure central interactions using a plane truss; the equivalent Born–Kármán's lattice in plane stress in the limit tends to a continuum with Poisson's ratio υ = 1 / 3 . Contrary to McHenry–Hrennikoff's truss, Born–Kármán's lattice leads to a "free" Poisson's ratio bounded by its "limit' bound ( υ = 1 / 4 for plane strain or 3D elasticity; υ = 1 / 3 for plane stress elasticity). Unfortunately, Born–Kármán's lattice model does not comply with rotational invariance principle, for non-central forces. The consistent generalization of Lagrange's lattice in 3D was achieved only by Gazis et al. considering an elastic energy that depends on changes in both lengths and angles of the lattice. An alternative consistent three-parameter elastic lattice is the Hrennikoff's, with additional structure in the cell. We also discuss the capability of nonlocal continuous models to bridge the gap between continuum isotropic elasticity at low frequencies and lattice anisotropic elasticity at high frequencies. [ABSTRACT FROM AUTHOR]

• The proposed integration scheme can accurately solve various orders of singularities under a unified framework. • The use of NURBS in the present integration scheme improves the accuracy of the singular integrals for complex boundaries. • The effects of various weight points and integration schemes in the time-weighting method on stability are discussed. The paper presents a novel numerical scheme to effectively solve the weak, strong, and hyper-singular time-space integrals in the 3D time domain boundary element method (BEM). To deal with the singularities of P- or S-wavefront elements, the element subdivision method is used to identify non-zero valued sub-elements on the wavefronts, and the non-zero valued sub-elements containing source points are solved by the power series expansion method, which achieves the effect of obtaining satisfactory accuracy with fewer Gaussian points. In the present scheme, the non-uniform rational B-Spline (NURBS) basis functions are used to determine the physical coordinates of each sub-element with very high precision. Numerical results show that the proposed scheme can effectively implement a high-precision singular time-space integral solution under a unified framework. In addition, this paper uses the time-weighting method to improve the time integration algorithm to solve the instability problem. The validity and accuracy of the algorithm are verified by dynamics examples of the finite elastic body and infinite elastic body, and the effects of different weight points and integration schemes in the time-weighting method on the stability are discussed. The effectiveness of the proposed algorithm for large-scale elastodynamics problems is also examined using the 27 spherical cavities problem. [ABSTRACT FROM AUTHOR]

The elastic moduli dispersion of materials is a topic of great interest in many engineering practices. Such dispersion is regularly measured on laboratory-preferred cylindrical samples of various natures, from solid dry to single-porosity and dual-porosity dual-permeability fluid-saturated materials. Analytical modeling of these laboratory tests is desired as it is effective and time-efficient in the interpretation of the experimental results and the characterization of the poromechanical properties of the material. To this end, for the first time, this paper presents analytical modeling of the elastic moduli dispersion and the poromechanical behaviors of a dual-porosity dual-permeability fluid-saturated cylinder under a dynamic forced deformation test. Our dual-porosity dual-permeability poroelastodynamics solution can be easily reduced to the elastodynamics and the single-porosity poroelastodynamics ones. We demonstrate the capabilities of our analytical solution by modeling the dynamic elastic moduli and poromechanical responses of a naturally fractured rock sample. Our results show that spatial distributions of poromechanical quantities, such as pore pressure and strain, are mostly uniform at low frequencies and become increasingly nonuniform at high frequencies. Additionally, the dynamic Young's modulus and Poisson's ratio are highly dependent on Biot's and Skempton's coefficients, the sample's dimension, and the loading frequency. The dispersion trends of these dynamic elastic moduli vary as the loading frequency approaches the material's resonant and anti-resonant frequencies. Finally, we use our analytical solution to successfully match laboratory data from dynamic forced deformation tests on three clastic sediment rock samples from the North Sea and a shale sample from Mont Terri. Highlights: The first analytical poroelastodynamics solution for a dual-porosity dual-permeability fluid-saturated porous cylinder. Comprehensively interpretation of the poromechanical responses and the mechanisms of the dispersion. Excellent matches between the analytical solution with laboratory measurements. [ABSTRACT FROM AUTHOR]

We report on a class of architected material lattices that exploit multi-stability and kinematic amplification to independently adjust the local effective mass, damping, and stiffness properties, thereby realizing congruent alterations to the acoustic dispersion response post-fabrication. The fundamental structural tuning element permits a broad range in the effective property space; moreover, its particular design carries the benefit of tuning without altering the original size/shape of the emerging structure. The relation between the tuning element geometry and the achieved variability in effective properties is explored. Bloch's theorem facilitates the dynamic analysis of representative one- and two-dimensional (1D/2D) systems, revealing, e.g., bandgap formation, migration, and closure and positive/negative metadamping in accordance with the tuning element configuration. To demonstrate a utility, we improvise a waveguide by appropriately patterning the tuning element configuration within a 2D system. We believe that the proposed strategy offers a new way to expand the range of performance and functionality of architected materials for elastodynamics. [ABSTRACT FROM AUTHOR]

It is well known that the kinetic and potential energies of a system governed by a linear wave equation are equipartitioned in a finite time. This result has been extended by previous researchers to several physical settings: elastodynamics, electrodynamics, and micropolar elasticity. In the case of a helix, there is a direct coupling of longitudinal and torsional wave motions due to the structure of constitutive relations. Providing that the initial conditions functions have compact support, we demonstrate that the equipartition phenomenon occurs after a finite time in this case as well—i.e. the potential energy and the kinetic energy of a helix become equal and remain constant after a finite time. [ABSTRACT FROM AUTHOR]