Discretization effects in the finite element simulation of seismic waves in elastic and elasticplastic media
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Abstract
Presented here is a numerical investigation that (re)appraises standard rules for space/time discretization in seismic wave propagation analyses. Although the issue is almost off the table of research, situations are often encountered where (established) discretization criteria are not observed and inaccurate results possibly obtained. In particular, a detailed analysis of discretization criteria is carried out for wave propagation through elastic and elasticplastic media. The establishment of such criteria is especially important when accurate prediction of highfrequency motion is needed and/or in the presence of highly nonlinear material models. Current discretization rules for wave problems in solids are critically assessed as a conditio sine qua non for improving verification/validation procedures in applied seismology and earthquake engineering. For this purpose, the propagation of shear waves through a 1D stack of 3D finite elements is considered, including the use of wideband input motions in combination with both linear elastic and nonlinear elasticplastic material models. The blind use of usual rules of thumb is shown to be sometimes debatable, and an effort is made to provide improved discretization criteria. Possible pitfalls of wave simulations are pointed out by highlighting the dependence of discretization effects on time duration, spatial location, material model and specific output variable considered.
Keywords
Wave propagation Seismic Discretization Elastic Elasticplastic Verification1 Introduction
The study of wave motion is of utmost importance in many applied sciences, as it supports the understanding of transient phenomena in many natural and anthropic dynamic systems. In particular, seismic waves propagating through the earth crust deserve the highest consideration, especially in light of their destructive potential and socioeconomical impact.
In the last decades, mathematicians, geophysicists and engineers have devoted massive research efforts to the prediction of seismic motion, based on either analytical [21, 32, 33, 34, 39] or numerical methods [2, 53, 63]. When linear elastic wave problems are considered, either timedomain or frequencydomain solutions may be sought, whereas timedomain approaches are usually needed in the presence of nonlinearities (constitutive or geometrical). In this respect, it should be remarked that much interest in earthquake engineering is nowadays on nonlinear wave phenomena, since they govern (i) the occurrence of natural catastrophes (e.g., landslides and debris flows) induced by soil instabilities, such as liquefaction and strain localization [18, 24, 63]; (ii) the interaction between geomaterials and manmade structures [13, 16, 20, 28, 53, 59].
 1.
selection of the numerical solution algorithm;
 2.
mathematical description of material behavior (constitutive model);
 3.
computer implementation;
 4.
setup of the computational discrete model.
The present work focuses on the fourth item in the list, and specifically on the selection of appropriate timestep and element size in dynamic Finite Element (FE) computations. This problem seems to have been solved quite long ago in the form of “rules of thumb” for space/time discretization [38, 41], so that not many works on the subject have been published ever since [4, 5, 14, 55]. Furthermore, the relationship between discretization and accuracy in wave simulations has been mainly investigated for linear elastic problems.

only 1D shear wave propagation tests are performed for a more straightforward interpretation of numerical results;

discretization effects have been illustrated in both time and frequency domains, and then quantified via modern misfit criteria formulated in the full timefrequency domain;

since discretization effects depend in general on the numerical algorithm adopted, a widespread FE approximation scheme has been here adopted;

the role of constitutive nonlinearity (plasticity) is discussed;

the whole study should be regarded as a numerical “falsification test” for the “rules of thumb” previously mentioned [38, 41].
2 FE modeling of 1D seismic wave propagation
Like in general 3D problems, the numerical analysis of 1D seismic wave propagation requires a suitable computational platform for (i) space/time discretization, (ii) material modeling and (iii) simulation under given initial/boundary conditions. The Real ESSI Simulator has been used here for these purposes.
The Real ESSI Simulator is a software, hardware and documentation system developed specifically for highfidelity, realistic modeling and simulation of earthquakesoil structureinteraction (ESSI). The Real ESSI program features a number of simple and advanced modeling features. For example, on the finite element side, available are solids elements (8, 20, 27, 827 node, dry and saturated bricks), structural elements (trusses, beams, shells), contact elements (frictional slip and gap, dry and saturated), isolator and dissipator elements; on the material modeling side, available are elastic (isotropic, anisotropic, linear and nonlinear) and elasticplastic models (isotropic, anisotropic hardening). The seismic input can be applied using the Domain Reduction Method [7, 61], while sequential and parallel versions of the program are available (the latter is based on the Plastic Domain Decomposition (PDD) method [25]). Recent applications of Real ESSI to seismic problems are documented, for instance, in [1, 12, 27, 28, 29, 30, 46, 56, 57, 58].
2.1 Space discretization and time marching
The Real ESSI program is based on a standard displacement FE formulation, where displacement components are taken as unknown variables in the numerical approximation [62]. As for space discretization, the 1D FE model has been built using a stack of properly constrained 3D brick elements—as was previously done, for instance, by [10]. Real ESSI program enables the use of 8, 20 and 27node elements, so that several options are given in terms of spatial interpolation order.
2.2 Material modeling
The Real ESSI program provides a number of material modeling options, ranging from simple linearelastic to advanced elasticplastic constitutive relationships for cyclically loaded soils [18, 63]. Hereafter, the material models adopted for wave propagation analyses are briefly described, namely (i) the standard linear elastic material model, (ii) the elasticplastic von Mises model with linear kinematic hardening [26, 40] and (iii) the bounding surface elasticplastic model by [48].
2.2.1 Linear elastic model
2.2.2 Elasticplastic: von Mises kinematic hardening (VMKH) model
The relationship among discretization, accuracy and material nonlinearity will be first explored through the elasticplastic von Mises kinematic hardening (VMKH) model, of the same kind described in [26, 40].

two elastic parameters—E and \(\nu\);

one yielding parameter—k—proportional to the initial size of the cylindrical yield locus in the stress space;

one hardening parameter—h—governing the postyielding (elasticplastic) stiffness.
2.2.3 Elasticplastic: Pisanò bounding surface (PBS) model
 1.
development of inelastic strains from the very onset of loading. This is reproduced by exploiting the concept of “vanishing yield locus”;
 2.
frictional shear strength, i.e., depending on the effective confining pressure;
 3.
nonlinear hardening, implying a continuous transition from smallstrain to failure stiffness;
 4.
coupling between deviatoric and volumetric responses;
 5.
stiffness degradation and damping under cyclic shear loading.

two elastic parameters—E and \(\nu\)—to characterize the material behavior at vanishing strains;

one shear strength parameter—M—directly related to the material frictional angle;

two parameters—\(k_d\) and \(\xi\)—governing the development of plastic volumetric strains during shearing;

two hardening parameters—h and m—to be identified on the basis of stiffness degradation and damping cyclic curves.
2.3 Initial/boundary conditions and input motion
 1.
the system is initially at rest (nil initial velocities and accelerations);
 2.
a horizontal xdisplacement time history is imposed at the bottom boundary to reproduce rigid bedrock conditions;
 3.
no loads are applied to the top boundary (free surface);
 4.
the aforementioned “double planestrain” conditions has been achieved by preventing ydisplacements throughout the model, as well as imposing master/slave connections to nodes at the same elevation (tied nodes).
The above features of the Ormsby wavelet will enable the analysis of discretization effects over frequency ranges of choice. Although most seismic energy relates to frequencies lower than 20 Hz, ensuring accuracy at higher frequencies may be relevant when seismic serviceability analyses are to be performed for structures, systems and components (SSCs) related to nuclear power plants and other industrial objects.
2.4 Misfit criteria
The analysis of discretization effects requires objective criteria to quantify the discrepancy (misfit) between different numerical solutions. In numerical seismology, the difference seismogram between the numerical solution and a reliable reference solution is often adopted for this purpose, although it only enables visual/qualitative observations; simple integral criteria (e.g., root mean square misfit) can provide some quantitative insight, but still with no distinction of amplitude or phase errors.
3 Linear elastic wave simulations
In this section, the influence of discretization on accuracy is first discussed for linear elastic problems.
3.1 Standard rules for space/time discretization
3.2 Model parameters
The geometrical/mechanical parameters adopted for elastic wave simulations are here reported. A uniform soil layer has been considered, having thickness H = 1 km and made of an elastic material with \(\rho =2000\) kg/m^{3}, \(V_\mathrm{s}=1000\) m/s and \(\nu =0.3\) (corresponding to \(G=2\) GPa). No Rayleigh damping has been introduced.

input 1: \(f_1\) = 0.1 Hz, \(f_2\) = 1 Hz, \(f_3\) = 18 Hz, \(f_4\) = 20 Hz (plotted in Fig. 2);

input 2: \(f_1\) = 0.1 Hz, \(f_2\) = 1 Hz, \(f_3\) = 45 Hz, \(f_4\) = 50 Hz;

the amplitude parameter A has been always set to produce at the bottom of the layer a maximum displacement of 1 mm.
3.3 Discussion of numerical results
The influence of grid spacing and timestep size are discussed separately for the sake of clarity. Since the Real ESSI program is based on a displacement FE formulation, displacement components are the most reliable output; however, some attention is also paid to accelerations, postcalculated through secondorder central differentiation.
List of elastic simulations
Case #  \(f_\mathrm{max}\) (Hz)  \(\Delta x_\mathrm{std}\) (m)  \(\Delta t_\mathrm{std}\) (s)  \(\Delta x\) (m)  \(\Delta t\) (s)  Brick type 

EL1  20  5  0.005  2, 5, 10  0.005  8node 
EL2  20  5  0.005  2, 5, 10  0.002  8node 
EL3  50  2  0.002  0.8, 2, 4  0.002  8node 
EL4  50  2  0.002  0.8, 2, 4  0.001  8node 
EL5  20  5  0.005  2, 5, 10  0.002  27node 
EL6  20  5  0.005  5  0.002, 0.005, 0.01  8node 
EL7  20  5  0.005  2  0.001, 0.002, 0.005  8node 
EL8  50  2  0.002  2  0.001, 0.002, 0.005  8node 
EL9  50  2  0.002  0.8  0.0005, 0.001, 0.002  8node 
EL10  20  5  0.005  5  0.002, 0.005, 0.01  27node 
The results being presented aim to assess the quality of standard discretization rules, as well as the improvements attainable through refined discretization. For this purpose, the numerical results are discussed in both time and frequency domains—the Fourier spectra of considered time histories are plotted in terms of (i) amplitude and (ii) phase difference with respect to the analytical solution (known at the free surface). Additional quantitative insight is also gained through the EM and PM misfit criteria introduced in Sect. 2.4. Unless differently stated, numerical outputs at the top of the soil layer are considered.
3.3.1 Influence of grid spacing

even though \(\Delta x_\mathrm{std}\) is set on the basis of the maximum frequency \(f_\mathrm{max}\), its suitability is not uniform over the input spectrum. Indeed, increasing inaccuracies in the frequency domain are clearly visible as \(f_\mathrm{max}\) is approached (check for instance the Fourier amplitudes compared in Figs. 3c and 4, 5, 6b). Grid spacing affects output Fourier spectra both in amplitude and phase;

in all cases, envelope and phase misfits, EM and PM, are quantitatively very similar (Figs. 3e and 4, 5, 6d);

reducing \(\Delta x\) below \(\Delta x_\mathrm{std}\) is beneficial only if \(\Delta t\) is also lower than \(\Delta t_\mathrm{std}\). This is apparent in Fig. 3e, where an increase in EM and PM is observed as \(\Delta x\) gets lower than \(\Delta x_\mathrm{std}\). Conversely, monotonic EM/PM trends are shown in Figs. 4, 5d;

at given grid spacing \(\Delta x\), reducing the timestep improves the numerical solution mostly in terms of Fourier phase, not amplitude (compares Figs. 3c–d, 4b–c). It may be generally stated that, when \(\Delta x\) is not appropriate, reducing the timestep size does not produce substantial improvements;

based on these initial examples, a grid spacing \(\Delta x\) in the order of \(V_\mathrm{s}/20f_\mathrm{max}=\Delta x_\mathrm{std}/2\) ensures high accuracy (EM and PM < 10 % ) in combination with \(\Delta t=\Delta x/2V_\mathrm{s}=\Delta t_\mathrm{std}/2\). These enhanced discretization rules hold for loworder FEs (8node brick elements) but are not affected by the frequency bandwidth of the input signal. In the latter respect, Figs. 4, 5d show quantitatively similar EMPM trends for \(f_\mathrm{max}\) equal to 20 Hz and 50 Hz. Also, minimum misfits are attained in the EL2 case (Fig. 4d), where a smaller \(\Delta t/\Delta t_\mathrm{std}\) ratio has been purposely set.
It is also important to evaluate grid spacing effects on acceleration components, as they will affect the inertial forces transmitted to manmade structures on the ground surface. Since acceleration time histories are dominated by high frequencies, the poorer performance of standard discretization rules at high frequencies becomes more evident. In Figs. 7 and 8, grid spacing plays qualitatively as in Figs. 3, 4, 5, although the EM/PM trends—similar in shape—are shifted upwards. This means that, in the presence of loworder elements, more severe discretization requirements should be fulfilled if very accurate accelerations are needed.
3.3.2 Influence of timestep size

as observed in the previous subsection, \(\Delta t\) mainly affects the Fourier phase, with comparable EM and PM values in all cases. Phase differences with respect to the exact solution decrease as \(\Delta t\) is reduced – see for instance in Figs. 9, 10, 11, 12c;

in combination with \(\Delta x=V_\mathrm{s}/20f_\mathrm{max}=\Delta x_\mathrm{std}/2\), \(\Delta t=\Delta t_\mathrm{std}\) may still result in some highfrequency phase difference with the respect to the analytical solution, (Figs. 9, 10, 11, 12c). As found by investigating grid spacing effects, \(\Delta t={\Delta x}/{2V_\mathrm{s}}=\Delta t_\mathrm{std}/2\) yields sufficient accuracy (EMPM lower than 10 %) to most practical purposes (see Figs. 9, 10, 11, 12d);

when 27node bricks are used, the use of \(\Delta x=\Delta x_\mathrm{std}\) and \(\Delta t\le \Delta t_\mathrm{std}/2\) is still an appropriate option, giving rise to EM and PM lower than 5 % (Fig. 12). Even in this case, discretization errors are still governed by phase differences, while excellent performance in terms of Fourier amplitude is observed;

Figs. 13 and 14 show that the above findings apply qualitative to acceleration time histories as well. However, EM and PM values are quite high (significantly larger than 10 %) when \(\Delta t\ge \Delta t_\mathrm{std}\), regardless of the grid spacing ratio. Accuracy is quickly regained when \(\Delta t\) is reduced and \(\Delta x<\Delta x_\mathrm{std}/2\).
While the above conclusions have been all drawn on the basis of the first incoming wave, many reflected waves may in reality hit the ground surface because of soil layering. In the present elastic case (no energy dissipation), perfect reflections occur at the lower rigid bedrock and neverending wave motion is established. It is thus interesting to check how discretization errors propagate in time at the free surface, as is shown in Fig. 15. Subsequent wave arrivals are compared in the time (Fig. 15a, b) and frequency (Fig. 15c, d) domains, where a gradual “accumulation” of wave dispersion can be observed. Even though satisfactory accuracy is achieved on the first arrival, an increase in highfrequency phase difference is detected in Fig. 15d, with negligible variation in Fourier amplitude (Fig. 15c). Cumulative wave dispersion implies that selecting suitable \(\Delta x\) and \(\Delta t\) becomes increasingly delicate for large FE models and long durations.
4 Nonlinear elasticplastic wave simulations

the nonlinear problem under consideration cannot be solved analytically. Therefore, the quality of discretization settings may only be assessed by evaluating the converging behavior of numerical solutions upon \(\Delta x\)–\(\Delta t\) refinement;

with no analytical solution at hand, one needs engineering judgement to establish when the (unknown) exact solution is reasonably approached. In this respect, light is shed on several expected pitfalls, all relevant to the global verification process [3, 45, 51];

the accuracy of nonlinear computations is highly affected by the input amplitude. This governs the amount of nonlinearity mobilized by wave motion and, as a consequence, the accuracy of numerical solutions at varying discretization.
4.1 VMKH model
4.1.1 Model parameters and parametric analysis

mass density and elastic properties: \(\rho =\) 2000 kg/m^{3}, \(E=\) 5.2 GPa and \(\nu =\) 0.3, whence the elastic shear wave velocity \(V_\mathrm{s}=\) 1000 m/s results (same elastic parameters employed for both the elastic and the VMKH sublayers);

yielding parameter (radius of the von Mises cylinder): \(k=\) 10.4 kPa;

different h values (hardening parameter) have been set: \(h=\) 0.5E, 0.05E, 0.01E.
List of VMKH simulations
Case #  \(\Delta x_\mathrm{std}\) (m)  \(\Delta t_\mathrm{std}\) (s)  \(\Delta x\) (m)  \(\Delta t\) (s)  h  A (mm) 

VMKH1  5  0.0005  1, 5  0.0001  0.5E  0.1 
VMKH2  5  0.0005  1, 5  0.0001  0.05E  0.1 
VMKH3  5  0.0005  5  0.0002, 0.0005, 0.001  0.5E  0.1 
VMKH4  5  0.0005  5  0.0002, 0.0005, 0.001  0.05E  0.1 
VMKH5  5  0.0005  5  0.0002, 0.0005, 0.001  0.01E  0.1 
VMKH6  5  0.0005  1, 5  0.0001  0.5E  1 
VMKH7  5  0.0005  1, 5  0.0001  0.05E  1 
VMKH8  5  0.0005  5  0.0002, 0.0005, 0.001  0.5E  1 
VMKH9  5  0.0005  5  0.0002, 0.0005, 0.001  0.05E  1 
VMKH10  5  0.0005  5  0.0002, 0.0005, 0.001  0.01E  1 
4.1.2 Influence of grid spacing and timestep size

propagation through a dissipative elasticplastic material alters significantly the shape of the input signal. All plots display significant wave attenuation/distortion, while final unrecoverable displacements are produced by soil plastifications (Figs. 16, 17a). Steady irreversible deformations are associated with prominent static components (at nil frequency) in the Fourier amplitude spectrum (Figs. 16, 17c), not present in the input Ormsby wavelet (Fig. 2b);

the numerical representation of wavelengths is dominated by soil plasticity, producing more deviation from the input waveform than variations in grid spacing. For this reason, only two \(\Delta x\) values have been used in this subsection for illustrative purposes, whereas EM/PM plots have been deemed not necessary;

the influence of \(\Delta x\) seems slightly magnified when lower h values, and thus lower elasticplastic stiffness, are set (see Fig. 17). It is indeed not surprising that wave propagation in softer media may be more affected by space discretization, as in linear problems. However, it should be noted that \(\Delta x\) mainly influences the final irreversible displacement (Fig. 17b, c), which leads to presume substantial interplay of grid effects and constitutive time integration;

since the effects of \(\Delta x\) reduction are quite small in both time and frequency domains (for a given \(\Delta t\)), there is no strong motivation to suggest \(\Delta x=V_\mathrm{s}/20f_\mathrm{max}\). \(\Delta x=V_\mathrm{s}/10f_\mathrm{max}=\Delta x_\mathrm{std}\) should be actually appropriate in common practical situations, as long as no soil failure mechanisms are triggered – as for example in seismic slope stability problems [17]. The occurrence of soil failure may introduce additional discretization requirements for an accurate representation of the collapse mechanism.

at variance with the previous elastic cases, envelope (EM) and phase (PM) misfits are quantitatively quite different (EM > PM);

EM/PM trends do not depend monotonically on the hardening parameter h. For \(\Delta t=\) 0.0002 s, the EM/PM values at \(h=0.05E\) are indeed larger than those obtained for \(h=0.5E\) and \(h=0.01E\).
4.1.3 Influence of input motion amplitude
In nonlinear problems, it is hard to draw general conclusions on the interaction between space/time discretization and input amplitude. The latter governs the amount of soil nonlinearity mobilized and the resulting local stiffness, in turn affecting the requirements for accurate constitutive integration.
In Fig. 21, the parametric study in Figs. 16, 17 is replicated for a higher input amplitude (\(A=\) 1 mm) and the same two different h values (cases VMKH67 in Table 2). The timedomain plots provided testify the effects of grid spacing on the predicted response: again, they mostly concern the final residual displacement, more pronouncedly as h decreases. The same previous uncertainties about the interplay of grid spacing and constitutive integration still apply to this case.
The discussion on the influence of \(\Delta t\) at higher input amplitude refers to Figs. 22, 23, illustrating the results obtained for \(\Delta x=\Delta x_\mathrm{std}\) and h equal to 0.5E, 0.05E and 0.01E (cases VMKH810 in Table 2); EM/PM plots comes from the numerical reference solution corresponding with \(\Delta t=\Delta t_\mathrm{std}/5=\) 0.0001 s.

\(\Delta t\) affects not only the residual component of displacement time histories (as in Fig. 21), but also their maximum/minimum transient values – i.e., the numerical representation of plastic dissipation. This is clearly visible in Fig. 22a;

EM/PM values are in general higher at larger input amplitude (Fig. 22d), and experience a slower decrease as \(\Delta t\) is reduced (still depending on the specific h value);

the shear stressstrain loops in Fig. 23 show how inaccurate the simulated constitutive response can be when \(\Delta t\) is too large (e.g., \(\Delta t=\) 0.001 s) and substantial plastic degradation of material stiffness takes place (see the case \(h=0.01E\)).
4.2 PBS model
4.2.1 Model parameters and parametric analysis
The influence of space/time discretization is now explored in combination with the nonlinear PBS soil model introduced in Sect. 2.2.3 [48]. As in real geomaterials, the PBS model features an elasticplastic response since the very onset of loading (vanishing yield locus), with the stiffness smoothly evolving from smallstrain elastic behavior to failure (nil stiffness).

\(\rho =\) 2000 kg/m^{3}, \(E=\) 1.3 GPa and \(\nu =\) 0.3, implying an elastic shear wave velocity \(V_\mathrm{s}=\) 500 m/s ;

shear strength parameter: \(M=\) 1.2, corresponding with friction angle equal to 30 deg under triaxial compression;

dilatancy parameters: \(k_d=\) 0.0 and \(\xi =\) 0.0^{4};

hardening parameters: \(h=\) 300 and \(m=\) 1.
List of PBS simulations
Case#  \(\Delta x_\mathrm{std}\) (m)  \(\Delta t_\mathrm{std}\) (s)  \(\Delta x\) (m)  \(\Delta t\) (s)  A (mm) 

PBS1  2.5  0.0005  0.5, 2.5  0.0001  1 
PBS2  2.5  0.0005  0.1, 0.5, 1  0.00002  1 
PBS3  2.5  0.0005  2.5  0.0002, 0.0005, 0.001  1 
PBS4  2.5  0.0005  2.5  0.00001, 0.00002, 0.0001  1 
4.2.2 Influence of grid spacing and timestep size

grid spacing turns out to be influential again (Figs. 24, 26), as a consequence of more severe variations (than in VMKH cases) in shear stiffness during cyclic loading. In fact, one would have to follow the stiffness reduction curves arising from the constitutive response, and use minimum stiffness to decide on space discretization;

as in VMKH simulations, grid spacing mainly affects residual displacements. This is clearly shown by the EM/PM plots in Fig. 26b, where EM errors larger than 10% arise even when a very small timestep size is used (\(\Delta t =\Delta t_\mathrm{std}/25=\) 0.00002 s); conversely, phase misfits are less affected by residual displacements and thus always quite limited. In presence of high nonlinearity, it seems safer to use \(\Delta x\) \(4\div 5\) times smaller than \(\Delta x_\mathrm{std}=V/10f_\mathrm{max}\);

the combination of explicit constitutive integration and high nonlinearity makes timestepping effects quite prominent, as is shown by Figs. 27 and 28. Further, Fig. 29 leads to conclude that \(\Delta t=\Delta t_\mathrm{std}/50\) may be needed to obtain EM errors lower than 10 % (Figs. 29, 30). Apparently, analysts have to compromise on accuracy and computational costs in these situations;

as expected, the shear stressstrain cycles in Figs. 25 and 28 show that the sensitivity to discretization builds up as increasing nonlinearity is mobilized. This is the case for instance at the top of the PBS layer, where cycles are more dissipative than at the bottom due to lower overburden stresses and dynamic amplification.
Since displacement components result from strains through spatial integration, the displacement performance can be wellpredicted on condition that strains are accurately computed all along the soil domain. For the same reason, the discretization requirements for displacement convergence are not uniform along the soil deposit. Figs. 31 and 32 illustrate in the timedomain the displacements simulated at different depths in the nonlinear sublayer (the vertical x axis points upward—Fig. 1) and at different \(\Delta x\) and \(\Delta t\). These figures clearly point out that accuracy requirements may be more or less hard to satisfy depending on the specific spatial location. In 1D wave propagation problems, faster convergence is attained far from the ground surface, since it requires satisfactory accuracy in a lower number of nodes and integration points.
5 Concluding remarks
Previously established criteria for space/time discretization in wave propagation FE simulations have been reappraised and critically discussed to strengthen verification procedures in Computational Dynamics. The 1D propagation of seismic shear waves (Ormsby wavelets) through both linear and nonlinear (elasticplastic) media has been numerically simulated, with focus on capturing highfrequency motion and exploring the relationship between material response and discretization effects. After initial linear computations, two different nonlinear material models (referred to as VMKH and PBS) have been used at increasing level of complexity. The main conclusions inferred are hereafter summarized:
Elastic simulations Setting grid spacing (element size) and timestep size as per standard rules (\(\Delta x_\mathrm{std}=V_\mathrm{s}/10f_\mathrm{max}\) and \(\Delta t_\mathrm{std}=\Delta t/V_\mathrm{s}\)) has proven not always appropriate, especially to reproduce highfrequency motion components (this can be clearly visualized in the Fourier phase plane). When linear elements (8node bricks) are used, \(\Delta x\approx \Delta x_\mathrm{std}/2\) and \(\Delta t\approx \Delta t_\mathrm{std}/2\) seem to ensure sufficient accuracy over the whole frequency range (both in amplitude and phase); higherorder elements (e.g., 27node bricks) will allow the use of \(\Delta x=\Delta x_\mathrm{std}\) still in combination with \(\Delta t\approx \Delta t_\mathrm{std}/2\). Preserving accuracy in simulations with large domains and/or time durations seems intrinsically more difficult, since attenuation/dispersion phenomena are cumulative.
Elasticplastic simulations Conclusive criteria for elasticplastic problems can be hardly established, as space/time discretization also interferes with the integration of nonlinear constitutive equations. In this respect, different outcomes may be found depending on (i) kind of nonlinearity associated with the material model (stiffness variations during straining), (ii) stresspoint integration algorithm (e.g., explicit or implicit), (iii) input motion amplitude. The experience gained through the use of the PBS model (explicitly integrated in 8node brick elements) suggests that \(\Delta x_\mathrm{std}=V_\mathrm{s}/10f_\mathrm{max}\) and \(\Delta t_\mathrm{std}=\Delta x/10V_\mathrm{s}\) may need to be reduced by factors up to \(4\div 5\) and 50, respectively, in the presence of strong input motions and severe stiffness variations. Importantly, these conclusions also depend on which output component is considered and where within the computational domain.
The present study is, however, not conclusive, especially when it comes to nonlinear elasticplastic problems. There are in fact several aspects that will deserve in the future further consideration, such as the implications of using higherorder finite elements. The same comment applies to geometrical effects (e.g., wave scattering) in 2D/3D problems, whose influence on discretization criteria for elasticplastic simulations would be per se a whole research topic.
Footnotes
 1.
 2.
Henceforth, \(\Delta x\) will always denote the vertical node spacing, coinciding with the element thickness in the case of 8node bricks.
 3.
For a given number of nodes per wavelength, the size \(\Delta x\) of 27node elements along the propagation direction is double than for 8node bricks.
 4.
Soil volume changes under shear loading have been inhibited for the sake of simplicity. This aspect would further affect the overall stiffness of the soil layer and require additional parametric analyses.
Notes
Acknowledgments
The Authors wish to acknowledge the financial support from the USNRC, USDOE and Shimizu Inc. Corporation (Japan).
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