Elastodynamic Equations Research Articles

Exponential time propagators for elastodynamics

We propose a computationally efficient and systematically convergent approach for elastodynamics simulations. We recast the second-order dynamical equation of elastodynamics into an equivalent first-order system of coupled equations, so as to express the solution in the form of a Magnus expansion. With any spatial discretization, it entails computing the exponential of a matrix acting upon a vector. We employ an adaptive Krylov subspace approach to inexpensively and accurately evaluate the action of the exponential matrix on a vector. In particular, we use an apriori error estimate to predict the optimal Krylov subspace size required for each time-step size. We show that the Magnus expansion truncated after its first term provides quadratic and superquadratic convergence in the time-step for nonlinear and linear elastodynamics, respectively. We demonstrate the accuracy and efficiency of the proposed method for one linear (linear cantilever beam) and three nonlinear (nonlinear cantilever beam, soft tissue elastomer, and hyperelastic rubber) benchmark systems. For a desired accuracy in energy, displacement, and velocity, our method allows for 10−100× larger time-steps than conventional time-marching schemes such as Newmark-β method. Computationally, it translates to a ∼1000× and ∼10−100× speed-up over conventional time-marching schemes for linear and nonlinear elastodynamics, respectively.

Numerical Study of Laminar Flow and Vortex-Induced Vibration on Cylinder Subjects to Free and Forced Oscillation at Low Reynolds Numbers

In this study, we aimed to numerically investigate the 2D laminar flow over a cylindrical body and performed vortex-induced vibration analyses on a circular cylinder of unit radius placed in a channel, with the cylinder assumed to be fixed. The cases of a cylinder under forced oscillation and three different scenarios of a freely oscillating cylinder were analyzed. The fluid domain dynamics were governed by the incompressible Navier–Stokes equations; however, the structural field was described using nonlinear elastodynamic equations. Fluid and solid domains were discretized with the finite volume method (FVM) in space and time. Predictions of hydrodynamic forces, namely lift and drag terms, were determined for each scenario. An increase in the Reynolds number caused an exponential increment in the lift force. In the case of a stabilized flow, the collective decrease in stiffness and damping decreased the maximal drag and lift factors. Furthermore, it was noticed that the lift factor was minimally altered by variations in damping and stiffness in comparison with the change in the drag factor. From these observations, it appears that the lift factor probably correlates with the cylinder’s structure and fluid properties.

Efficient Guided Wave Modelling for Tomographic Corrosion Mapping via One-Way Wavefield Extrapolation

Mapping corrosion depths along pipeline sections using guided-wave-based tomographic methods is a challenging task. Accurate defect sizing depends heavily on the precision of the forward model in guided wave tomography. This model is fitted to measured data using inversion techniques. This study evaluates the effectiveness of a recursive extrapolation scheme for tomography applications and full waveform inversion. It employs a table-driven approach, with precomputed extrapolation operators stored across a spectrum of wavenumbers. This enables fast modelling for extensive pipe sections, approaching the speed of ray tracing while accurately handling complex velocity models within the full frequency band. This ensures an accurate representation of diffraction phenomena. The study examines the assumptions underlying the extrapolation approach, namely, the negligible reflection and conversion of modes at defects. In our tomography approach, we intend to use multiple wave modes—A0, S0, and SH1—and helical paths. The acoustic extrapolation method is validated through numerical studies for different wave modes, solving the 3D elastodynamic wave equation. Comparison with an experimentally measured single-mode wavefield from an aluminium plate with an artificial defect reveals good agreement.

Natural frequencies and modal shapes of folded sandwich plates made of porous core and FG-CNTRC coating layers resting on two parameters elastic foundation

Folded structures hold significant importance in aerospace technology due to their distinct design, effectively reconciling strength with a lightweight framework. Within aerospace applications, these structures fulfill diverse roles, serving as pivotal elements in aircraft wings, fuselage sections, and essential components of spacecraft and satellites. This study delves into the free vibration behavior of folded sandwich plates composed of a porous core and functionally graded carbon nanotube-reinforced composite (FG-CNTRC) coatings, resting on a two-parameter elastic foundation. Leveraging first-order shear deformation plate theory (FSDT), elasto-dynamic equations are established for each individual plate segment. Utilizing the two-dimensional generalized differential quadrature method (2D-GDQM), these equations are discretized. Subsequently, by applying boundary conditions to all six edges, including traditional clamped, simply supported, and free configurations, and incorporating continuity conditions for displacements, forces, and moments at the joint borders, natural frequencies and modal shapes are extracted for the folded plate structure. The validity of the proposed formulation and results is demonstrated by comparing them with numerical examples for both composite flat and folded plates. This comparison provides strong support for the accuracy and reliability of the present study. The study comprehensively examines the influence of various parameters on natural frequencies, including modal shape visualizations under different boundary configurations, plate thickness, crank angle, porosity parameter, core thickness, foundation stiffness, and CNTs distribution type.

Efficient testing of weld seam models with radii of curvature in the millimeter range using laser ultrasound

Laser ultrasound is a widely used tool for industrial quality assurance when a contactless and fast method is required. In this work, we used a laboratory setup based on a confocal Fabry–Perot interferometer to examine weld seam models. The focus was placed on small samples with curved surfaces (small in the sense that the radius of curvature is comparable to the largest ultrasonic wavelength) and on efficient ways to detect the presence and volume of process pores, with the goal to transfer this method to industrial applications. In addition to this experimental method for investigating welds, a numerical method that models the experimental setup was implemented in MATLAB. For this purpose, first the thermal effects of the excitation process were taken into account by solving the thermal diffusion equation with an explicit scheme. Then, the elastodynamic equations were solved using the Elastodynamic Finite Integration Technique, taking into account the stresses induced by the excitation process. The B-Scans generated with this numerical model were compared with experimental B-Scans for simple test cases and good agreement was found. In a next step, the additional structures in the B-Scans resulting from air inclusions were identified and investigated with both methods using flat test specimens at first. Besides the direct echoes, structures from skimming surface waves and multiple reflections were visible. These additional structures are unwanted in defect reconstruction methods like the Synthetic Aperture Focusing Technique (SAFT) as they would lead to artifacts. In samples much larger than the largest ultrasound wavelength, however, these unwanted structures are still negligible in amplitude or can be well separated temporally, but for small samples this is no longer the case. As a result, reconstruction methods based on direct echoes like SAFT are difficult to apply. For many industrial applications, the reconstruction is not decisive at all, but only the knowledge of the total volume of process pores (TVPP). It is shown with both experimental and numerical methods, that this TVPP can be estimated from the variation in the B-Scans from various small weld seam models.

3D elastodynamic analysis of transversely isotropic laminated water-saturated subsoils under harmonically moving loads

This article proposes an analytical layer element methodology for transversely isotropic (TI) laminated water-saturated subsoils to a harmonically moving load over the media surface. Biot’s elastodynamic equation is decoupled by the double Fourier transform combined with the Heaviside step function of the load source. Exact 3D dynamical stiffness matrices of the soil layer element and the TI saturated half-space are established based on the derived displacement and stress components in the transform domain. Assembling the stiffness matrices of the discrete layer elements and underlying half-space yields the total 3D dynamical stiffness matrices. Considering the element boundary conditions, the analytical layer-element solutions in the wavenumber domain are obtained. The space domain result is retrieved by the inverse Fourier transformation. The results have a good consistency with the current solutions. The effects of the soft soil layer thickness, the material anisotropy, stratification characteristics, the load frequency, and the load moving velocity are analyzed.

The 3D Physics-Based Broadband Ground-Motion Simulation, including Topography Effects Causing Damage to Structures, for the 2022 Mw 6.6 Menyuan, China, Earthquake

ABSTRACT On 8 January 2022, Menyuan County, China, was struck by an Mw 6.6 earthquake that caused surface rupture in the epicentral area and severe damage to an important railway bridge. The earthquake was recorded by only one strong-motion station, which presents a challenge for quantitatively estimating the extent of the ground-motion distribution caused by this event. In this study, the spectral element method (SPECFEM3D code), which solves the elastodynamic equations and can capture the full physics of seismic-wave propagation, is employed for broadband (0–10 Hz) ground-motion simulations of this earthquake. A hybrid kinematic source is developed in which the final slip distribution combines a prescribed asperity model based on Interferometric Synthetic Aperture Radar data (as a source for low-frequency radiation) and a stochastic part (as a source for high-frequency radiation), which introduces spatial heterogeneities to the prescribed asperity model. The numerical approach is first validated by modeling the well-recorded 1994 Northridge earthquake before modeling the Menyuan earthquake. The simulated ground motion is compared with the only observed strong-motion station record, as well as with empirical Next Generation Attenuation-West2 ground-motion models. Then, topography effect in Menyuan earthquake is studied in detail. The simulated ground motions with and without surface topography indicate that the topography tends to focus and scatter the seismic wavefield, resulting in amplification of the ground shaking. The results show a significant correlation between the peak ground velocity (PGV) and topography. The PGV amplification caused by topography effects is period dependent, and its peak amplification reaches up to 50% within a typical resonance period (1–2 s). It could be inferred that the railway bridge probably vibrated in resonance and suffered severe damage owing to the amplified long-period ground motion caused by the topography.

Physics-informed deep learning for scattered full wavefield reconstruction from a sparse set of sensor data for impact diagnosis in structural health monitoring

This paper presents a physics-informed deep learning framework for the reconstruction of full scattered spatiotemporal Lamb wavefields (video images) in plate-like structures from a sparse set of time-series sensor data. The reconstructed scattered wavefield contains a wealth of information about the wave propagation phenomenon including any interactions of the propagating wave with damage in the structure. This information is paramount for damage diagnosis as is demonstrated in this paper via impact diagnosis—a key structural health monitoring application. A physics-informed neural network (PINN) that encodes the underlying elastodynamic field equations into the learning/training process in the neural network is proposed for this purpose. This prior wavefield physics knowledge embedded in the loss function acts as a regularization agent for the minimization problem in the neural network training, thereby enabling the extrapolation of a sparse set of one-dimensional time-series signals into two-dimensional scattered wavefield. The wavefield reconstruction capabilities of the proposed supervised forward PINN framework are first verified both numerically and experimentally for a stiffened aluminum panel under a couple of narrowband ultrasonic-frequency excitations, and the results confirm its robustness to low spatial resolution and substantial noise in the measured sensor data. The PINN requires far fewer sensors for scattered wavefield reconstruction, thereby permitting for a higher sensor spacing or lower spatial sampling. To this end, it is shown that a sensor spacing of 5λ generates good wavefield reconstruction accuracy, which is a 10-fold increase over the Nyquist–Shannon sampling limit (λ/2). Two sets of experiments are then conducted on a long-stiffened aluminum panel to validate the proposed framework via low-velocity impact diagnosis in the near-ultrasonic frequency range. The first set of experiments, with the known excitation force incorporated into the PINN, allows the wavefields to be accurately reconstructed with the sensor spacing up to 5λ as expected. The second set of experiments assumes unknown impact force history—a classical case of impact diagnosis where the impact force history is not known a priori. It is shown that the wavefield reconstruction through PINN still provides good accuracy albeit with a less generous sensor spacing of 2λ. A convolutional neural network long short-term memory (CNN-LSTM) model then solves the mathematically inverse problem of inferring the impact location and impact force history by analyzing the reconstructed impact generated wavefield. The impact location is predicted well with 93% accuracy, and the impact force history is reconstructed with 90% accuracy, further validating the proposed framework.

Empirical scaling of formation fracturing by high-energy impulsive mechanical loads

Recent technological advances to trigger high-energy seismic waves from within the wellbore have spurred interest in their potential application to induce fracturing within deep subsurface formations. Although a considerable body of experimental observations at the bench scale (on the order of 1 cubic foot) show promise, there remains considerable uncertainty in how the process may scale to the field. This work characterizes the relationship between the extent and intensity of fracturing and the duration and strength of the dynamic load. Our approach uses a direct numerical simulation of the elastodynamic equations while accounting for nonlinear fracture mechanics. A hybrid Finite-Discrete Element Method (FDEM) is applied whereby cohesive (elastoplastic) laws hold mesh elements together until complete failure. Beyond failure, elements act as deformable free bodies that can interact via contact constraints, including friction. The consistency of the model is first verified and it is validated with observations from previously reported physical experiments. Subsequently, we apply the model to conduct a numerical exploration of the dimensionless parameter space that defines plausible loading waveforms. We model an infinite domain that contains a spherical inclusion within which various impulsive loads are applied. The dynamic loading waveform is parameterized by its rise time to peak stress, followed by a decay period, and occurring within micro- to milliseconds. The dimensionless parameter space is sampled by varying loading characteristics (rise time, peak pressure, and impulse) to reveal the stimulated damaged bulk volume and crack intensity. Consistent with reported experimental observations, the model reproduces a nearly linear trend between the radius of damage and the maximum stress on the bench scale. Beyond that scale, however, the model predicts that this scaling slows considerably to a fractional power law between the damaged radius and the peak stress. This limitation coincides with a geometric increase in the intensity of damage within the stimulated volume. We present loci curves of the expected damage radii as functions of the loading waveform properties.

Scattering of elastic waves by a sphere with orthorhombic anisotropy and application to polycrystalline material characterization

Scattering of elastic waves by an anisotropic sphere with orthorhombic symmetry inside an isotropic medium is studied and applied to characterization of polycrystalline materials with anisotropic grains. For a single sphere the waves in the isotropic surrounding are expanded in the spherical vector wave functions. Inside the sphere, the elastodynamic equations are first transformed to spherical coordinates and the displacement field is expanded in terms of the vector spherical harmonics in the angular directions and power series in the radial direction. The governing equations inside the sphere give recursion relations among the expansion coefficients in the power series. The boundary conditions on the sphere then determine the relation among the scattered wave expansion coefficients and those of the incident wave, expressed as the transition (T) matrix. For low frequencies the elements of the T matrix are obtained in explicit form. According to the theory of Foldy the T matrix elements of a single sphere are used to study attenuation and phase velocity of polycrystalline materials, explicitly for low frequencies. Comparisons of the present method with previously published results and recent FEM results show a good correspondence for low frequencies. The present approach shows a better agreement with FEM for strongly anisotropic materials in comparison with other published methods.

An implicit–explicit time discretization for elastic wave propagation problems in plates

AbstractWe propose a new implicit–explicit scheme to address the challenge of modeling wave propagation within thin structures using the time‐domain finite element method. Compared to standard explicit schemes, our approach renders a time marching algorithm with a time step independent of the plate thickness and its associated discretization parameters (mesh step and order of approximation). Relying on the standard three dimensional elastodynamics equations, our strategy can be applied to any type of material, either isotropic or anisotropic, with or without discontinuities in the thickness direction. Upon the assumption of an extruded mesh of the plate‐like geometry, we show that the linear system to be solved at each time step is partially lumped thus efficiently treated. We provide numerical evidence of an adequate convergence behavior, similar to a reference solution obtained using the well‐known leapfrog scheme. Further numerical investigations show significant speed up factors compared to the same reference scheme, proving the efficiency of our approach for the configurations of interest.

A damped elastodynamics system under the global injectivity condition: local wellposedness in L^p-spaces

The purpose of this paper is to model mathematically mechanical aspects of cardiac tissues. The latter constitute an elastic domain whose total volume remains constant. The time deformation of the heart tissue is modeled with the elastodynamics equations dealing with the displacement field as main unknown. These equations are coupled with a pressure whose variations characterize the heart beat. This pressure variable corresponds to a Lagrange multiplier associated with the so-called global injectivity condition. We derive the corresponding coupled system with nonhomogeneous boundary conditions where the pressure variable appears. For mathematical convenience a damping term is added, and for a given class of strain energies we prove the existence of local-in-time solutions in the context of the L^p-parabolic maximal regularity.

Physics-informed neural networks for approximating dynamic (hyperbolic) PDEs of second order in time: Error analysis and algorithms

We consider the approximation of a class of dynamic partial differential equations (PDEs) of second order in time by the physics-informed neural network (PINN) approach, and provide an error analysis of PINN for the wave equation, the nonlinear Klein-Gordon equation and the linear elastodynamic equation. Our analyses show that, with feed-forward neural networks having two hidden layers and the tanh activation function, the PINN approximation errors for the solution field, its time derivative and its gradient field can be effectively bounded by the training loss and the number of training data points (quadrature points). Our analyses further suggest new forms for the training loss function, which contain certain residuals that are crucial to the error estimate but would be absent from the canonical PINN loss formulation. Adopting these new forms for the loss function leads to a variant PINN algorithm. We present ample numerical experiments with the new PINN algorithm for the wave equation, the Sine-Gordon equation and the linear elastodynamic equation, which show that the method can capture the solution well.

A study of deep neural networks for Newtonian noise subtraction at Terziet in Limburg—the Euregio Meuse-Rhine candidate site for Einstein Telescope

The Euregio Meuse-Rhine border region of Belgium, Germany and the Netherlands has been identified as a candidate site for hosting Einstein Telescope. Newtonian coupling of ground vibrations to the core optics of the detectors may limit the sensitivity of Einstein Telescope at frequencies below about 10 Hz. The contribution of Newtonian noise is site specific and depends on the ambient seismic field which in turn depends on the site’s geology and the distribution of surface and underground seismic-noise sources. We have investigated the application of machine learning in combination with the deployment of seismic sensor networks to predict seismic displacement noise at specific locations on the surface and underground. Moreover we have modeled a deep neural network (DNN) that allows to subtract Newtonian noise from the strain data measured by Einstein Telescope. The seismic-field model is based on a complete solution of the elastodynamic wave equations for a horizontally-layered soil structure. The geology features soft-soil layers on hard-rock and was shown to be effective in attenuating Newtonian noise from surface waves below the required sensitivity. In addition our model includes a random background of body waves with all possible angles of incidence. We show that a DNN is effective in predicting Newtonian noise. Application of our DNN allows Newtonian noise subtraction by a factor up to 4.7 at 1 Hz and 2.5 at 5 Hz.

Observation of Brillouin scattering in a high-index doped silica chip waveguide

We report the observation of Brillouin backscattering in a 50-cm long spiral high-index doped silica chip waveguide and measured a Brillouin frequency shift of 16 GHz which is in very good agreement with theoretical predictions and numerical simulations based on the elastodynamics equation.© 2023.

Time domain boundary elements for elastodynamic contact

This article proposes a boundary element method for the dynamic contact between a linearly elastic body and a rigid obstacle. The Signorini contact problem is formulated as a variational inequality for the Poincaré-Steklov operator for the elastodynamic equations on the boundary, which is solved in a mixed formulation using boundary elements in the time domain. We obtain an a priori estimate for the resulting Galerkin approximations. Numerical experiments confirm the stability and convergence of the proposed method for the contact problem in flat and curved two-dimensional geometries, as well as for moving obstacles.

Soil–pile interaction in vertical vibration in inhomogeneous soils

AbstractThis paper develops a novel reference analytical solution for axially loaded piles in inhomogeneous soils, extending the pioneering elastodynamic model of Nogami and Novak (1976) to piles embedded in vertically inhomogeneous soils. Following the classical earlier model, the pile is modelled as a rod, using the strength‐of‐materials solution, and the soil layer as an approximate continuum, which rest on rigid rock. The approximation lies in reducing the number of dependent variables by eliminating certain stresses and displacements in the governing elastodynamic equations: the vertical normal and vertical shear stresses in the soil are controlled exclusively by the vertical component of the soil displacement. Soil inhomogeneity is introduced via a power law variation of shear modulus with depth, and perfect bonding is assumed at the soil–pile interface. The proposed generalized formulation treats two types of inhomogeneity by employing pertinent eigen expansions of the dependent variables over the vertical coordinate. The response is expressed in terms of generalized Fourier series and includes: (i) displacements and stresses along the pile and the pile–soil interface; and (ii) displacement and stress in the soil. Contrary to available models for homogeneous soils, the associated Fourier coefficients are coupled, obtained as solutions to a set of simultaneous algebraic equations of equal rank to the number of modes considered.

Computing dispersion diagrams and forced responses of arbitrarily varying waveguides

Periodic structures can be used for vibration control thanks to their vibration attenuation stop bands. The dispersion diagram obtained for the structural unit cell can be used to characterize such wave attenuation bands. Several designs using spatially varying unit cells have recently been reported in the literature. Spatially varying properties can also be used to investigate the effect of variability on finite structures. Therefore, various methodologies to model spatially varying structures have been recently developed. This work presents new numerical methodologies to compute (i) dispersion diagrams and (ii) forced responses of one-dimensional spatially varying periodic waveguides. The first method extends previous work using a state-space formulation to compute dispersion relations using a linear time-varying system dynamics approach. The methodology is applied to elementary rods, Saint-Venant shafts, Euler–Bernoulli beams, and Timoshenko beams, and the results are validated with discretized spectral element models and the plane wave expansion method. The proposed approach is shown to be more efficient in cases where the plane wave expansion method presents a slow convergence. The second method is a recursive dynamic condensation to compute forced responses using the dynamic stiffness matrix of the unit cell. The results and computational times of the proposed recursive condensation are compared with the direct assembly of a global dynamic stiffness and its standard dynamic condensation. The two proposed methods are shown to be more efficient and accurate in the computation of dispersion diagrams and forced responses of one-dimensional structures for which the elastodynamic equations are known.

A Mixed‐Flux‐Based Nodal Discontinuous Galerkin Method for 3D Dynamic Rupture Modeling

AbstractNumerical simulation of rupture dynamics provides critical insights for understanding earthquake physics, while the complex geometry and heterogeneous material properties of natural faults make numerical method development challenging. The discontinuous Galerkin (DG) method is suitable for handling these complexities. In the DG method, the fault boundary conditions can be conveniently imposed through the upwind flux by solving a Riemann problem based on a velocity‐strain elastodynamic equation. However, the universal adoption of upwind flux can cause spatial oscillations in cases where elements on adjacent sides of the fault surface are not nearly symmetric. Here we propose a nodal DG method with an upwind/central mixed‐flux scheme to solve the spatial oscillation problem and thus reduce the dependence on mesh quality. Our method can achieve high scalability in parallel computing under different orders of accuracy. We verify the new method by comparing simulation results with those from other methods on a series of published benchmark problems with complex fault geometries, heterogeneous materials, off‐fault plasticity, roughness, thermal pressurization, and various versions of fault friction laws. Finally, we demonstrate that our method can be applied to simulate the dynamic rupture process of the 2008 Mw 7.9 Wenchuan earthquake along realistic fault geometry, including branches, step‐overs and a previously neglected sub‐parallel fault segment, offering a high potential to systematically explore the influences of fault zone properties on earthquake rupture dynamics.

Natural frequency prediction of the 3-RPS parallel manipulator using the substructure synthesis technology

AbstractThis paper proposed an elastodynamic modeling method combined with independent displacement coordinates and substructure synthesis technology. Firstly, the connecting rod was discretized, and the elastodynamic control equation for each element was established. The multipoint constraint element theory, linear algebra, and singularity analysis were used to identify the globally independent displacement coordinates of the manipulator. On this basis, the elastodynamic model using the substructure synthesis for the 3-PRS parallel manipulator (PM) was developed, with its natural frequencies distribution in the regular workspace discussed. The comparison with the finite-element results showed that the maximum error of the first three-order natural frequencies was within 1.03%, which verified the correctness of the analytical model. The proposed elastodynamic model included all the kinematic constraints of the manipulator without increasing the Lagrangian multiplier. The method is computationally efficient and assesses the dynamic behaviors of the mechanism at the predesign phase.