Elastodynamic Problems Research Articles

The simplified weak Galerkin method with θ scheme and its reduced-order model for the elastodynamic problem on polygonal mesh

This paper presents a simplified weak Galerkin (SWG) method for solving the elastodynamic problem and its reduced-order model (ROM) using the proper orthogonal decomposition (POD) technique. The SWG method allows for the use of polygonal meshes. It only utilizes degrees of freedom associated with the boundary, reducing computational complexity compared to the classical weak Galerkin method. Moreover, we apply the POD technique to develop a POD-SWG-ROM for the problem, further enhancing the computational efficiency. Then, to discretize in time, we utilize a θ-scheme, where the scheme is explicit when 0≤θ<1/4 and implicit when 1/4≤θ≤1/2. We establish the theoretical analysis of the semi-discrete scheme and the fully-discrete θ scheme. The theoretical analysis demonstrates that the method is locking-free, and the convergence rate in the H1 and L2 norms is O(Δt2+h1) and O(Δt2+h2) respectively. Finally, we verify the theoretical analysis through numerical tests and effectively simulate the propagation of elastic waves under polygonal meshes. Moreover, the proposed POD-SWG-ROM can significantly improve computational efficiency.

Approximation of acoustic black holes with finite element mixed formulations and artificial neural network correction terms

Wave propagation in elastodynamic problems in solids often requires fine computational meshes. In this work we propose to combine stabilized finite element methods (FEM) with an artificial neural network (ANN) correction term to solve such problems on coarse meshes. Irreducible and mixed velocity–stress formulations for the linear elasticity problem in the frequency domain are first presented and discretized using a variational multiscale FEM. A non-linear ANN correction term is then designed to be added to the FEM algebraic matrix system and produce accurate solutions when solving elastodynamics on coarse meshes. As a case study we consider acoustic black holes (ABHs) on structural elements with high aspect ratios such as beams and plates. ABHs are traps for flexural waves based on reducing the structural thickness according to a power-law profile at the end of a beam, or within a two-dimensional circular indentation in a plate. For the ABH to function properly, the thickness at the termination/center must be very small, which demands very fine computational meshes. The proposed strategy combining the stabilized FEM with the ANN correction allows us to accurately simulate the response of ABHs on coarse meshes for values of the ABH order and residual thickness outside the training test, as well as for different excitation frequencies.

Wave solutions in nonlocal integral beams

Wave propagation in slender beams is addressed in the framework of nonlocal continuum mechanics. The elastodynamic problem is formulated exploiting consistent methodologies of pure integral, mixture and nonlocal strain gradient elasticity. Relevant wave solutions are analytically provided, with peculiar attention to reflection and near field phenomena occurring in presence of boundaries. Notably, the solution field is got as superimposition of incident, reflected, primary near field and secondary near field waves. The latter contribution represents a further effect due to the size dependent mechanical behaviour. Limit responses for vanishing nonlocal parameter are analytically evaluated, consistently showing a zero amplitude of the secondary near field wave. Parametric analyses are carried out to show how length scale parameter, amplitude of incident wave and geometric and elastic properties of the beam affect the amplitudes of reflected, primary near field and secondary near field waves. The results obtained exploiting different nonlocal integral elasticity approaches are compared and discussed.

An efficient PGD solver for structural dynamics applications

We propose in this paper a Proper Generalized Decomposition (PGD) solver for reduced-order modeling of linear elastodynamic problems. It primarily focuses on enhancing the computational efficiency of a previously introduced PGD solver based on the Hamiltonian formalism. The novelty of this work lies in the implementation of a solver that is halfway between Modal Decomposition and the conventional PGD framework, so as to accelerate the fixed-point iteration algorithm. Additional procedures such that Aitken’s delta-squared process and mode-orthogonalization are incorporated to ensure convergence and stability of the algorithm. Numerical results regarding the ROM accuracy, time complexity, and scalability are provided to demonstrate the performance of the new solver when applied to dynamic simulation of a three-dimensional structure.

A level set based topology optimization for elastodynamic problems using BEM

The paper presents a topology optimization methodology for 2D elastodynamic problems using the boundary element method (BEM). The topological derivative is derived based on the variation method and the adjoint variable method. The level set method is employed for the representation of the material domain and voids within a specified design domain. Thus, the boundaries can easily be generated, following the zero isocontour of the level set function. Numerical implementation is carried out to demonstrate the effectiveness of the proposed topology optimization methodology in wave isolation and waveguide problems.

Identification of interfacial strength between carbon fiber and epoxy matrix in CFRP laminates using pulse laser spallation method

Carbon fiber reinforced plastics (CFRP) have high specific strength and specific stiffness and are used in various fields such as aircraft, space structures, automobiles, and other transportation equipment. Since the internal structure of CFRP is very complex, it is very difficult to define failure criteria when external loads are applied. Various methods have been proposed to evaluate the interfacial shear strength between carbon fiber and resin matrix, such as microbond test, rupture test, and pull-out test, but currently no effective method has been proposed to evaluate the tensile strength of the CF/resin interface. In this study, we applied the pulsed laser spallation method, which has been confirmed to be effective for evaluating the adhesion strength of thin films and coating films on substrates, to the evaluation of CFRP. A finite element method based on the Laplace transform was used for the numerical analysis of the unsteady elastodynamic problem. The macroscopic stresses generated by ultrasonic waves excited by pulsed laser irradiation were calculated by inverse analysis using a transfer function with the displacement history of the back surface of the CFRP specimen as supplementary information. Microscopic stresses at the fiber/resin interface were evaluated by the homogenization method, taking into account the effects of fiber arrangement, distance between fibers, and molding residual stress in unidirectionally reinforced CFRP laminates. As a result, this series of investigations revealed that the vertical interfacial stress is dominant with respect to the strength of the fiber/resin interface with the strength of about 54MPa.

Elastodynamic shape derivative: focus on the sampling of a circular distribution

The detection and quantification of defects plays a crucial role in several industries. Assessing the severity enables to schedule maintenance appropriately, optimizing costs and reducing risks. Guided elastic waves are particularly suitable for Structural Health Monitoring applications on thin structures thanks to their long range and high sensitivity. Guided wave tomography, based on an acoustic propagation hypothesis, has been studied over the last decade to quantitatively image defects in waveguides. The acoustic approximation of this family of methods allows fast computation and good estimation of defects for simple geometries such as plates or pipes. However, this hypothesis induces that a single guided mode is considered, without mode conversion, which limits its use to defects larger than two wavelengths in structures close to ideal waveguides. Hence the need of new imaging algorithms for smaller defects in less ideal structures such as pipes with welded supports. We present here an adaptation of the shape derivative method to ultrasounds in order to reconstruct surface defect on structures with complex geometries. The physical model used is the full elastodynamic problem. It allows taking into account all physical phenomena occuring during wave propagation. Iterative methods such as shape derivative are well known to give precise results but at a high computational cost as the method needs to solve the full elastodynamic problem at each iteration, and a sensitivity to local minima, which may prevent convergence toward the true defect. To suppress these two drawbacks, a spectral finite element method is used to reduce the computation and memory costs. The result of acoustic guided wave tomography is also used as a first guess to reduce the number of iterations and the sensitivity to local minima. After explaining the principle of the shape derivative, validations on simulated and experimental data will be presented in this talk.

Damage identification technique by model enrichment for structural elastodynamic problems

Structural Health Monitoring (SHM) techniques are key to monitor the health state of engineering structures, where damage type, location and severity are to be estimated by applying sophisticated techniques to signals measured by sensors. However, very localized damage detection algorithms applied to dynamics problems when dealing with rigid structures at low-frequency range remains still a big challenge. The last due to the low influence of very localized damage on the overall response of the structure (Saint-Venant principle). In this context, in the present work, we propose a methodology for locally correcting the models from collected data for elastodynamics problems at low-frequency range which is able to predict very localized damage. The proposed technique consists in enriching the structural model in a sparse way and solving the identification problem in the frequency domain, where the influence of damage over a large frequency band is exploited to improve the prediction of the damage location. The advantages and potential of the proposed technique are illustrated for the damage detection in a plate problem, demonstrating the advantages of the method in detecting very localized damage. The proposed technique is limited to a methodological description, and further developments should be considered to approach its applicability in an industrial scenario.

Development of governing partial differential equations of reinforcing thin films

In this paper, a novel interface model for elastic isotropic thin films perfectly bonded to the surrounding elastic media is introduced. Considering an average stress and displacement through the thin film thickness, the shear stress discontinuity across the interface is expressed in terms of spatial and time derivatives of displacement components. The derived interfacial equations are presented in a general form. As such, they are applicable to asymmetric elastodynamic problems of anisotropic solids with thin film interfaces. Employing the introduced interface model and Hankel transform technique, static Green's functions of an elastic isotropic half-space reinforced by a buried thin film are obtained. Results of some special cases, including axisymmetric loading, inextensible membrane, unreinforced half-space, and surface-coated half-space, are also presented. The robustness and effectiveness of the proposed interface model are shown by two solved numerical examples, and some elastic responses are plotted to show the impact of thin film stiffness. According to the numerical results, modeling elastic thin films as inextensible membranes can significantly alter the elastic responses.

Zonal free element method for free and forced vibration analysis of two- and three-dimensional structures

This paper presents a new numerical method called the zonal free element method (ZFREM) for the free and forced vibration analysis of elastodynamic problems. In this approach, a complex computational domain is divided into some simple zones and generates a series of regularly arranged nodes in each zone, which can improve the accuracy during the analysis of complex models. The distinguishing feature of the ZFREM is that an independent isoparametric element is formed by only one freely chosen surrounding node at each configuration node. In this method, the mass term exists only in the internal nodes, which can accelerate the assembly of the final system equations. Building upon this foundation, the present study developed the Krylov reduced dimensional iterative method, which approximates the solution of the equation system by constructing a lower-dimensional subspace. This approach avoids the complex equation transformations involved in traditional algorithms for solving eigenvalue problems, thereby further enhancing computational efficiency. Moreover, to tackle damped vibration problems encountered in engineering applications, the proposed method is further extended to solve the non-linear forced vibration problems. The accuracy and effectiveness of the method are verified by numerical examples of free and forced vibration problems.

Spectro-hierarchical homogenization scheme for elasto-dynamic problems in periodic Cauchy materials

A spectro-hierarchical algorithm is proposed to determine an approximate solution to the elasto-dynamic problem for periodic heterogeneous materials. Such a homogenization scheme is derived by employing tools from perturbation theory in finite dimension used in the context of the celebrated Theorem of Nekhoroshev. In particular, it is shown how a classical algorithm based on a suitable hierarchy of harmonics can be implemented for the problem at hand, leading to the explicit construction of functions approximating the solution of the original problem with an error that is superexponentially small in the cell dimension. According to this approach, the fully homogenized model turns out to be naturally related to the “integrable case” of perturbation theory. Furthermore, all the featured constants are estimated explicitly. More importantly, a fully detailed bound of the threshold for the cell dimension is presented to ensure the validity of the theory. An example of application of the described theory is given in the final section for the case of a layered material.

Non-locality as a regularization mechanism in elastodynamics

It is well known that the solutions to the elastodynamic problem do not satisfy continuous dependence properties on the initial values, and/or supply terms, when the elastic tensor fails to be positive. In fact, the behavior of the solutions can be very explosive since the elements of the spectrum can go to infinite. Therefore, it is very relevant to identify thermomechanical mechanisms regularizing the behavior of the solutions. So, the main aim of this note is to show, from an analytical point of view, how the non-locality, in the sense of Eringen, is a mechanism satisfying this property of regularization of the solutions. It is worth noting that such system has not been previously studied from an analytical point of view. We firstly obtain the existence of the solutions to this problem, even when we do not assume any positivity on the elastic tensor. This result is proved with the help of the linear semigroups theory; however, even with these regularizing effects, the solutions to this problem are unstable. A particular easy one-dimensional problem is also considered. The extension of the existence and instability results to the thermoelastic case is pointed out later. Finally, we also study the spatial behavior of the solutions to the problem in the case that the region is a semi-infinite cylinder, and we obtain a Phragmen–Lindelöf alternative of the exponential type. This result is also relevant because a similar result, without considering regularizing terms, is unknown if the elastic tensor is not positive definite.

Advanced absorbing boundaries for elastodynamic finite element analysis: The added degree of freedom method

Absorbing boundaries are essential in engineering simulations, especially for elastodynamic problems, to ensure accuracy, reliability and numerical stability. In this paper, the added degree of freedom method (ADM) is developed as a novel approach to address absorbing boundary challenges in finite element analysis. In ADM, additional degrees of freedom (DOFs) are introduced within the absorbing domain to attenuate outgoing elastic waves. For the proposed ADM to implement the absorbing boundary, the stiffness and mass properties of both the added DOFs and conventional DOFs within the absorbing domain are adjusted. This adjustment aims to reduce the propagation speed of elastic waves within the medium, and to prolong the duration of interaction between elastic waves and the surrounding medium. Consequently, the vibrations of nodes can be effectively attenuated by applying damping forces to the added DOFs. The numerical results show that the ADM has the ability to absorb one-dimensional and two-dimensional elastic waves across a broad range of frequencies. Therefore, the proposed ADM offers an innovative solution for modeling absorbing boundaries in various scientific and engineering applications, addressing the challenges of simulating wave propagation within finite computational domains.

POD method for solving elastodynamics problems of functionally gradient materials based on radial integration boundary element method

Functionally graded materials (FGMs) are widely utilized due to their excellent mechanical properties, prompting a need to explore the dynamic response of FGMs. To accurately and efficiently analyze elastodynamics problems, the radial integration boundary element method (RIBEM) is coupled with the proper orthogonal decomposition (POD) method in this paper. The POD method can transform a high-dimensional model into a low-dimensional system with high precision. Using the basic solution (Kelvin solution), the governing partial differential equation for the linear elastic dynamics problem of FGMs is transformed into a boundary-domain integral equation. The radial integration method is employed to convert the domain integral arising from material inhomogeneity and inertia terms into an equivalent boundary integral. This establishes a solution for the elastic dynamics problem of FGMs without the need for internal grid RIBEM. The Newmark time integration scheme is applied to solve the second-order ordinary differential equations. The displacement field obtained by RIBEM is utilized to construct the snapshots matrix, from which the POD reduced-order model is established. Numerical examples validate that the POD method significantly reduces the model's order, enhancing computational efficiency while also demonstrating the high accuracy of the POD method.

Structure-Preserving Discretization of a Polyconvexity-Inspired Formulation for Coupled Nonlinear Electro-Thermo-Elastodynamics

A consistent, structure-preserving space-time discretization for coupled nonlinear electro-thermo-elastodynamical problems is presented. The underlying polyconvexity-inspired mixed framework is facilitated by the properties of the tensor cross product. The elastodynamic problem is then extended by the energy balance as well as Gauss’s and Faraday’s law to integrate the thermodynamic and electrostatic contribution, respectively. A suitable polyconvexity-inspired internal energy function is chosen to complete the nonlinear, fully coupled electro-thermo-elastodynamical formulation. Additionally, we present a structure-preserving, second-order accurate time integration scheme, utilizing discrete derivatives in the sense of Gonzalez (1996), ensuring a stable and robust simulation even for large time steps. Finally, we assess the numerical performance of our newly developed method through representative examples also showing the possibilities of the framework in the field of boundary control.

Semi and fully discrete error analysis for elastodynamic interface problems using immersed finite element methods

Semi and fully discrete error analysis for elastodynamic interface problems using immersed finite element methods

Laplace-domain BEM for dynamic analysis of size-dependent couple stress elasticity problems

The size-dependent elastodynamic plane problems are described by a consistent couple-stress theory (C-CST) in this paper. The time-domain basic equations are then transformed to frequency domain by the Laplace transform method. The Laplace-domain boundary integral equations (BIEs) together with the fundamental solutions are derived. Then, these BIEs are numerically solved by a collocation method in conjunction with the numerical treatment of singular integrals. The time-dependent solutions are obtained by the Durbin's method for the inverse Laplace-transform. Some typical size-dependent elastodynamic problems are numerically studied by the developed boundary element method (BEM). Good agreements are observed between the present results and the corresponding analytical or numerical solutions, which validates the high accuracy of the present BEM.

Dynamic nucleation as a cascade-up of earthquakes depending on rupture propagation velocity

Earthquake dynamic rupture requires a nucleation process to provide sufficient energy to overcome the fracture energy. Large earthquakes may occur via a cascading rupture process, which includes many triggering processes that cascade from small to large sections of the fault system. During such a process, the nucleation of a large section of the fault plane may occur dynamically via the propagating rupture from a small section of the fault plane. A quasi-static view of seismic nucleation has been widely discussed in earthquake seismology; however, the dynamic nucleation process remains poorly known. Here, we investigate one aspect of the dynamic nucleation process by focusing on the rupture propagation velocity during the nucleation process. We simplify this process as self-similar crack propagation at a constant rupture velocity in a finite nucleation zone within a target region that possesses a uniform fracture energy. We numerically solve this elastodynamic problem in two dimensions for both the anti-plane and in-plane cases using the Boundary Integral Equation method. As the rupture velocity increases, the critical ratio of the fracture energy step to continue the rupture increases and the critical size of the dynamic nucleation zone decreases. The rapid increase in the ratio of the fracture energy step toward infinity could explain why earthquakes never propagate at slow rupture velocities. However, the effect on the size of the nucleation zone is rather limited, with the size of the dynamic nucleation zone decreasing to ~ 70% of the static nucleation zone size. However, such a small difference would result in a significant overall difference if such a dynamic nucleation process repeatedly occurred in the cascading rupture process of a large earthquake, which would be a difficult situation for earthquake early warning.Graphical

Frequency‐domain fundamental solutions and boundary element method for consistent couple stress elastodynamic problems

AbstractIn this article, the elastodynamic problems are investigated within the framework of a consistent couple‐stress theory (CCST). The basic equations for the CCST elastodynamic problems are derived at first. These dynamic formulas are then analyzed in the frequency domain by the Fourier transform. The frequency‐domain fundamental solutions to the transformed elastodynamic problems are subsequently derived for the two‐dimensional (2D) plane‐strain state. Based on the fundamental solutions and the elastodynamic reciprocal theorem, the frequency‐domain boundary integral equations (BIEs) are established. Then, the BIEs are numerically solved by a collocation method in conjunction with analytical and numerical treatments of the arising singular integrals. To obtain the time‐domain dynamic responses, an exponential window method (EWM) is adopted to transform the frequency‐domain results to the time‐domain solutions. By using the developed boundary element method (BEM), several typical couple‐stress elastodynamic problems are numerically investigated and the size‐effects are studied by using different values of the length‐scale parameter in the CCST. Good agreements between the results obtained by the developed BEM for a tiny length‐scale parameter and the corresponding classical BEM results or analytical solutions are achieved, which validates the high accuracy of the present BEM.

Solving elastodynamics via physics-informed neural network frequency domain method

Despite the fact that physics-informed neural networks (PINN) have been developed rapidly in recent years, their inherent spectral bias makes it difficult to approximate multi-frequency target functions such as the solutions to elastodynamics problems. To address this challenge, this paper proposes a new PINN frequency domain (PINNFD) method on the basis of frequency domain inputs. Fourier features for all frequencies can thus be constructed and embedded in the model, which provides essential support for the PINNFD method to accurately approximate multi-frequency target functions. To validate the effectiveness of the proposed PINNFD method, the proposed method and the traditional method are applied to solve the elastodynamics problem in infinite media under various dynamic point loads, including single-frequency harmonic load, multi-frequency harmonic load, and multi-frequency random load. The results show that although the PINN with embedded Fourier features is able to achieve the simulation of elastodynamics problems under harmonic loads, it fails to solve the problems under multi-frequency random loads. Whereas the proposed PINNFD method achieves better results in all cases of loading conditions, which demonstrates the advancement of the proposed method in solving multi-frequency problems in engineering applications.