Critical Time Step Research Articles

ER-MACG: An Extreme Precipitation Forecasting Model Integrating Self-Attention Based on FY4A Satellite Data

Extreme precipitation events often present significant risks to human life and property, making their accurate prediction an essential focus of current research. Recent studies have primarily concentrated on exploring the formation mechanisms of extreme precipitation. Existing prediction methods do not adequately account for the combined terrain and atmospheric effects, resulting in shortcomings in extreme precipitation forecasting accuracy. Additionally, the satellite data resolution used in prior studies fails to precisely capture nuanced details of abrupt changes in extreme precipitation. To address these shortcomings, this study introduces an innovative approach for accurately predicting extreme precipitation: the multimodal attention ConvLSTM-GAN for extreme rainfall nowcasting (ER-MACG). This model employs high-resolution Fengyun-4A(FY4A) satellite precipitation products, as well as terrain and atmospheric datasets as inputs. The ER-MACG model enhances the ConvLSTM-GAN framework by optimizing the generator structure with an attention module to improve the focus on critical areas and time steps. This model can alleviate the problem of information loss in the spatial–temporal convolutional long short-term memory network (ConvLSTM) and, compared with the standard ConvLSTM-GAN model, can better handle the detailed changes in time and space in extreme precipitation events to achieve more refined predictions. The main findings include the following: (a) The ER-MACG model demonstrated significantly greater predictive accuracy and overall performance than other existing approaches. (b) The exclusive consideration of DEM and LPW data did not significantly enhance the ability to predict extreme precipitation events in Zhejiang Province. (c) The ER-MACG model significantly improved in identifying and predicting extreme precipitation events of different intensity levels.

An automatic simulation pipeline for coupled simulations of acoustic damping materials

AbstractFoamed materials are widely used to reduce noise due to their comparably good acoustic damping behavior. However, out of a large variety of these materials a suitable candidate has to be identified for each application. This is a challenging process that is typically guided by experiments and experience. While numerical simulations could support these experiments and reduce the effort to a great extent, no suitable discretization approach has yet been established that can fully capture the complex geometry of the foam. A fully resolved model is desirable in order to yield reliable predictions that can then be used to establish homogenized models. We established a monolithic coupling approach based on the finite cell method (FCM) that realizes a vibroacoustic simulation in this sense. The fluid and the structure domain are discretized by Cartesian grids and the geometry defined based on computed tomography scans is accounted for during the quadrature of the weak form. Our simulation in the time domain makes use of explicit time marching schemes and is therefore limited by a critical time step size. This is known to be arbitrarily low for discretizations with the FCM containing cells with arbitrarily small support. As a remedy against this we use the classical ‐stabilization technique and investigate its potentials and limitations.

Unifying the contact in signed distance field-based and conventional discrete element methods

The signed distance field-based discrete element method (SDF-DEM) has demonstrated significant success in various applications; however, a key challenge hindering its widespread adoption lies in the establishment of its connection with conventional contact models. To address this challenge, this study introduces two formulations of contact potential within SDF-DEM, drawing analogies to both linear and Hertzian contact models. Comprehensive relationship between the parameters of the proposed contact potentials of SDF-DEM and those of conventional contact models is established. The energy conservation characteristic of SDF-DEM is verified through a two-particle colliding and bouncing test, and the critical timestep issue is investigated and addressed. Discrete element simulations are conducted for a triaxial compression test and a rockfall test, involving both spherical particles and general irregularly shaped particles. The results underscore the accuracy and numerical stability of the SDF-DEM with the developed potential models. This work is anticipated to contribute not only to advancing the understanding of SDF-DEM and the potential-based contact theory but also to providing robust framework that bridges the gap of SDF-DEM with conventional models.

Seasonal cap dynamics at the South Pole of Mars in the 36th Martian year, observed by Tianwen-1

The seasonal variation of the Martian South Polar Seasonal Cap (SPSC) is a significant factor to influence Mars atmospheric cycle, which has been continuously observed in previous missions. The SPSC degrades asymmetrically throughout the Martian seasons. However, developing a general description for this asymmetric recession is challenging, and this is crucial for identifying key stages (timesteps) which represents moments of special attribute throughout the recession. Here, we present the Martian Year 36 (MY36) results from the continuous observation of the Martian SPSC by the Moderate Resolution Imaging Camera (MoRIC) onboard the Tianwen-1 orbiter. A time-series of digital orthophoto maps (DOMs) characterizing the south cap were collected ranging from the solar longitude (Ls) 210° to 332°. A seasonal recession model of the cap was established by incorporating a 5th-order polynomial fitting of the time-series on cap latitude, from which the first derivative (recession rate), second derivative (acceleration), and third derivative (jerk) can be obtained. This methodology enables the extraction of critical timesteps that facilitate the categorization of the recession into distinct phases, each characterized by its recession rate and other factors. By the visual interpretation of remote sensing images obtained by Tianwen-1 and the result of timesteps showed that the pause time of the South Polar Residual Cap (SPRC) is about Ls = 324.5°-328° with an area of 7.9 × 104 km2. The discrepancies between our results and previous investigations indicate that the Martian climate presents inter-annual fluctuations. The MoRIC image data can be used to quantitatively capture the fine recession dynamic of the southern polar cap.The continuous operation of the Tianwen-1 orbiter will enable further data analysis for the dynamics of the Martian polar caps. This will provide observational constraints for Martian climate models and crucial information for the research of the Martian modern climate.

An analysis of high order FEM and IGA for explicit dynamics: Mass lumping and immersed boundaries

SummaryWe investigate the behavior of different shape functions for the discretization of hyperbolic problems. In particular, we consider classical Lagrange polynomials and B‐splines. The studies focus on the performance of the these functions as a spatial discretization approach combined with an explicit time marching scheme. In this regard, a major concern is the maximum eigenvalue that imposes restrictions on the critical time step size and suitable lumping techniques that yield a diagonal mass matrix. The accuracy of the discretization methods is assessed in an asymptotic manner in terms of the convergence of eigenvalues and eigenvectors. Further, the global accuracy is investigated in terms of the full spectrum. The results show that B‐spline discretization with a consistent mass matrix are more accurate than those based on Lagrange shape functions, which holds true in the boundary‐fitted as well as in the immersed setting. On the other hand, Lagrange shape functions are more robust with respect to standard lumping techniques, which cannot be directly applied for B‐splines without loss of accuracy. In general, we observe that none of the standard lumping schemes yields optimal results for B‐splines, even in the boundary‐fitted setting. For the immersed setting, also Lagrange shape functions show a drop in accuracy which depends on the position of the boundary that cuts the element. Several remedies are considered in order to overcome these issues, including interpolatory B‐spline bases as well as eigenvalue stabilization methods. While accuracy and stability can be improved using these remedies, we conclude from our study that it is still an open question, how to design a discretization method that achieves large critical time step sizes in combination with a diagonal mass matrix and high accuracy in the immersed setting. We note that these considerations primarily relate to linear structural dynamics applications, such as for example, structural acoustics. In nonlinear problems, such as automotive crash dynamics, other considerations predominate. An example of a one‐dimensional elastic‐plastic bar impacting a rigid wall is illustrative.

An eigenvalue stabilization technique for immersed boundary finite element methods in explicit dynamics

The application of immersed boundary methods in static analyses is often impeded by poorly cut elements (small cut elements problem), leading to ill-conditioned linear systems of equations and stability problems. While these concerns may not be paramount in explicit dynamics, a substantial reduction in the critical time step size based on the smallest volume fraction χ of a cut element is observed. This reduction can be so drastic that it renders explicit time integration schemes impractical. To tackle this challenge, we propose the use of a dedicated eigenvalue stabilization (EVS) technique.The EVS-technique serves a dual purpose. Beyond merely improving the condition number of system matrices, it plays a pivotal role in extending the critical time increment, effectively broadening the stability region in explicit dynamics. As a result, our approach enables robust and efficient analyses of high-frequency transient problems using immersed boundary methods. A key advantage of the stabilization method lies in the fact that only element-level operations are required.This is accomplished by computing all eigenvalues of the element matrices and subsequently introducing a stabilization term that mitigates the adverse effects of cutting. Notably, the stabilization of the mass matrix Mc of cut elements – especially for high polynomial orders p of the shape functions – leads to a significant raise in the critical time step size Δtcr.To demonstrate the efficiency of our technique, we present two specifically selected dynamic benchmark examples related to wave propagation analysis, where an explicit time integration scheme must be employed to leverage the increase in the critical time step size.

Numerical modelling of thermal jet assisted rock cutting with double PDC cutters

AbstractPreconditioning of rock for drilling operations is a potential method to facilitate the mechanical breakage and mitigate the tool wear. This paper numerically investigates one such preconditioning technique, namely, the thermal jet assisted rock cutting. For this end, a numerical method for solving the governing thermo‐mechanical problem is developed and validated. The continuum approach is chosen to describe the rock failure, being based on a damage‐viscoplasticity model with the Drucker–Prager yield surface and the modified Rankine surface as the tensile cut‐off. In the damage part, separate scalar damage variables are employed for tension and compression. The cutters are modelled as rigid bodies and their interaction with the rock is modelled by imposing contact constraints by the forward increment Lagrange multiplier method, which is compatible with the chosen explicit time integration scheme. A damage‐based erosion criterion is applied to remove the contact nodes surrounded by heavily damaged elements. The thermal jets are modelled with a moving external heat flux boundary condition. The global thermo‐mechanical problem is solved with a staggered approach explicitly in time while applying mass scaling to increase the critical time step. The novel 3D numerical simulations involving two cutters demonstrate the capabilities of the method with a special emphasis on cutter‐thermal jet configurations. Therefore, the present method provides a potential tool for the bit design in thermal jet assisted drilling.

An explicit material point and finite volume sequentially coupled method for simulating large deformation problems in saturated soil

An explicit material point and finite volume sequentially coupled method (MPM-FVM) is developed using u-p form coupling formulations for the simulation of large deformations in saturated soil. In MPM-FVM, the momentum conservation equation is solved by MPM under the updated Lagrange framework, and the mass conservation equations are solved by FVM under the Euler framework. The numerical implementation of an explicit sequentially coupled algorithm is developed considering the tradeoffs between computational cost and numerical stability. The critical time step of u-p form governing equations is derived using the Von Neumann method, which is found to be much greater than the u-U form explicit MPM at low permeability, allowing for high numerical stability. Three numerical cases are presented to demonstrate the stability, efficiency, and effectiveness of the developed MPM-FVM method. The simulation of a saturated one-dimensional soil column subjected to quasi-static and dynamic loads validates the method, and highlights its greater efficiency and accuracy compared to standard solid–fluid coupled MPM methods. The large deformation simulation capability of the method is exhibited via simulation of a saturated slumping block. In real-world application, simulation results obtained using method achieves excellent agreement with the experiment measurements and observations of the Selborne landslide experiment.

Implicit-explicit time integration for the immersed wave equation

Efficient solution strategies for wave propagation problems with complex geometries are essential in many engineering fields. Immersed boundary methods simplify mesh generation by embedding the domain of interest into an extended domain that is easy to mesh, introducing the challenge of dealing with cells that intersect the domain boundary. We consider the finite cell method that extends the weak form from the physical into the extended domain, multiplied by a small value, to stabilize badly cut cells. Combined with explicit time integration methods, the presence of finite cells with very little support in the physical domain results in tiny critical time step sizes. While the finite cell stabilization limits how small the critical time step size can become, this limit is still restrictive for many applications. Explicit transient analyses commonly use the spectral element method due to its natural way of obtaining diagonal mass matrices through nodal lumping. The resulting method is very efficient since nodal lumping renders the solution of the equation systems trivial while not reducing the accuracy or the critical time step size. The combination of the spectral element and finite cell methods is called the spectral cell method. Unfortunately, a direct application of nodal lumping in the spectral cell method is impossible due to the special quadrature necessary to treat the discontinuous integrand inside the cut cells. The existing approaches to lump the mass matrices of cut cells significantly reduce the accuracy of the approximation.We analyze an implicit-explicit (IMEX) time integration method to exploit the advantages of the nodal lumping scheme for uncut cells on one side and the unconditional stability of implicit time integration schemes for cut cells on the other. In this hybrid, immersed Newmark IMEX approach, we use explicit second-order central differences to integrate the uncut degrees of freedom that lead to a diagonal block in the mass matrix and an implicit trapezoidal Newmark method to integrate the remaining degrees of freedom (those supported by at least one cut cell). The immersed Newmark IMEX approach preserves the high-order convergence rates and the geometric flexibility of the finite cell method while retaining the efficiency of the nodal lumping scheme for uncut cells without compromising the critical time step size. We analyze a simple system of spring-coupled masses to highlight some of the essential characteristics of Newmark IMEX time integration. We then solve the scalar wave equation with immersed boundaries on two- and three-dimensional examples with significant geometric complexity to show that our approach is more efficient than state-of-the-art time integration schemes when comparing accuracy and runtime. While we focus on the finite and spectral cell methods, we expect other immersed methods to benefit equally from Newmark IMEX time integration.

Higher‐order inverse mass matrices for the explicit transient analysis of heterogeneous solids

AbstractNew methods are presented for the direct computation of higher‐order inverse mass matrices (also called reciprocal mass matrices) that are used for explicit transient finite element analysis. The motivation of this work lies in the need of having appropriate sparse inverse mass matrices, which present the same structure as the consistent mass matrix, preserve the total mass, predict suitable frequency spectrum and dictate sufficiently large critical time step sizes. For an efficient evaluation of the reciprocal mass matrix, the projection matrix should be diagonal. This condition can be satisfied by adopting dual shape functions for the momentum field, generated from the same shape functions used for the displacement field. A theoretically consistent derivation of the inverse mass matrix is based on the three‐field Hamilton principle and requires the projection matrix to be evaluated from the integral of these shape functions. Unfortunately, for higher‐order FE shape functions and serendipity FE elements, the projection matrix is not positive definitive and can not be employed. Therefore, we study several lumping procedures for higher order reciprocal mass matrices considering their effect on total‐mass preserving, frequency spectra and accuracy in explicit transient simulations. The article closes with several numerical examples showing suitability of the direct inverse mass matrix in dynamics.

A Hu-Washizu variational approach to self-stabilized quadrilateral Virtual Elements: 2D linear elastodynamics

A recent mixed formulation of the Virtual Element Method in 2D elastostatics, based on the Hu-Washizu variational principle, is here extended to 2D elastodynamics. The independent modeling of the strain field, allowed by the mixed formulation, is exploited to derive first order quadrilateral Virtual Elements (VEs) not requiring a stabilization (namely, self-stabilized VEs), in contrast to the standard VEs, where an artificial stabilization is always required for first order quads. Lumped mass matrices are derived using a novel approach, based on an integration scheme that makes use of nodal values only, preserving the correct mass in the case of rigid-body modes. In the case of implicit time integration, it is shown how the combination of a self-stabilized stiffness matrix with a self-stabilized lumped mass matrix can produce excellent performances both in the compressible and quasi-incompressible regimes with almost negligible sensitivity to element distortion. Finally, in the case of explicit dynamics, the performances of the different types of derived VEs are analyzed in terms of their critical time-step size.

Thermal and structural analysis of the European water-cooled lithium lead breeding blanket concept in the L-H mode transition

The design of a breeding blanket system (BB) which reliably fulfils the prescribed requirements is pivotal for the construction of the EU DEMO fusion reactor. For this reason, the BB conceptual design activities require a tighter interaction with plasma physics modelling, so to employ relevant boundary conditions to compare, and then down-select, the BB design alternatives. The preliminary thermo-mechanical study herein reported highlights the importance to assess in deep the BB performance during plasma operational transients. In particular, analyses of the BB thermal and mechanical transients during plasma ramp-up have been carried out, with focus on the L-H transition. Attention has been paid to the Water-Cooled Lithium Lead (WCLL) BB concept, one of the two driver blanket candidates for the EU DEMO project. Adopting DEMO-relevant plasma configurations, the fusion power and the radiative power time evolutions during the plasma ramp-up phase have been modelled under different assumptions using the code ASTRA. The obtained power transients have been used to calculate the time-dependent nuclear power density and radiative heat flux the WCLL BB is subjected to, in order to perform the corresponding transient thermal analysis. Then, the structural analysis has been carried out in order to verify the fulfilment of the RCC-MRx structural design criteria in the most critical time steps. The performed thermo-mechanical analysis has been carried out adopting the Abaqus FEM code. Results obtained are presented and critically discussed, highlighting their importance for the EU DEMO integrated design.

Prediction of Particulate Matter 2.5 Concentration Using a Deep Learning Model with Time-Frequency Domain Information

In recent years, deep learning models have gained significant traction and found extensive applications in the realm of PM2.5 concentration prediction. PM2.5 concentration sequences are rich in frequency information; however, existing PM2.5 concentration prediction models lack the ability to capture the frequency information. Therefore, we propose the Time-frequency domain, Bidirectional Long Short-Term Memory (BiLSTM), and attention (TF-BiLSTM-attention) model. First, the model uses Discrete Cosine Transform (DCT) to convert the time domain information into its corresponding frequency domain representation. Second, it joins the time domain information with the frequency domain information, which enables the model to capture the frequency domain information on top of the original. Simultaneously, incorporating the attention mechanism after BiLSTM enhances the importance of critical time steps. Empirical results underscore the superior predictive performance of our proposed univariate model across all sites, outperforming both the univariate BiLSTM, univariate BiLSTM-attention, and univariate TF-BiLSTM. Meanwhile, for the multivariate model that adds PM2.5 concentration from other sites in the study area as input variables, our proposed model outperforms the prediction of some basic models such as BiLSTM and some hybrid models such as CNN-BiLSTM for all sites.

Enstatite meteorite clasts in Almahata Sitta and other polymict ureilites: Implications for the formation of asteroid 2008 TC3 and the history of enstatite meteorite parent asteroids

AbstractThe anomalous polymict ureilite Almahata Sitta (AhS) fell in 2008 when asteroid 2008 TC3 disintegrated over Sudan and formed a strewn field of disaggregated clasts of various ureilitic and chondritic types. We studied the petrology and oxygen isotope compositions of enstatite meteorite samples from the University of Khartoum (UoK) collection of AhS. In addition, we describe the first bona fide (3.5 mm‐sized) clast of an enstatite chondrite (EC) in a typical polymict ureilite, Northwest Africa (NWA) 10657. We evaluate whether 2008 TC3 and typical polymict ureilites have a common origin, and examine implications for the history of enstatite meteorite asteroids in the solar system. Based on mineralogy, mineral compositions, and textures, the seven AhS EC clasts studied comprise one EHa3 (S151), one ELb3 (AhS 1002), two EHb4‐5 (AhS 2012, AhS 26), two EHb5‐6 or possibly impact melt rocks (AhS 609, AhS 41), and one ELb6‐7 (AhS 17), while the EC clast in NWA 10657 is EHa3. Oxygen isotope compositions analyzed for five of these are similar to those of EC from non‐UoK collections of AhS, and within the range of individual EC meteorites. There are no correlations of oxygen isotope composition with chemical group or subgroup. The EC clasts from the UoK collection show the same large range of types as those from non‐UoK collections of AhS. The enstatite achondrite, AhS 60, is a unique type (not known as an individual meteorite) that has also been found among non‐UoK AhS samples. EC are the most abundant non‐ureilitic clasts in AhS but previously were thought to be absent in typical polymict ureilites, necessitating a distinct origin for AhS. The discovery of an EC in NWA 10657 changes this. We argue that the types of materials in AhS and typical polymict ureilites are essentially similar, indicating a common origin. We elaborate on a model in which AhS and typical polymict ureilites formed in the same regolith on a ureilitic daughter body. Most non‐ureilitic clasts are remnants of impactors implanted at ~50–60 Myr after CAI. Differences in abundances can be explained by the stochastic nature of impactor addition. There is no significant difference between the chemical/petrologic types of EC in polymict ureilites and individual EC meteorites. This implies that fragments of the same populations of EC parent bodies were available as impactors at ~50–60 Myr after CAI and recently. This can be explained if materials excavated from various depths on EC bodies at ~50–60 Myr after CAI were reassembled into mixed layers, leaving relatively large bodies intact to survive 4 billion years. Polymict ureilites record a critical timestep in the collisional and dynamical evolution of the solar system, showing that asteroids that may have accreted at distant locations had migrated to within proximity of one another by 50–60 Myr after CAI, and providing constraints on the dynamical processes that could have caused such migrations.

An efficient detailed layer model for prediction of separator damage in a Li-Ion pouch cell exposed to transverse compression

Li-ion pouch cells can be understood as thin-layer laminates. Pressure loads normal to flat electrodes can cause failure or thermal runaway. However, detailed models for investigating transverse pressure loads in which each layer is resolved are computationally intensive. This paper outlines two approaches for time-efficient yet accurate detailed explicit finite element models of a cell. Both approaches are characterised by the fact that they are divided into in-plane and out-of-plane behaviour. The modelling of interlaminar contacts is completely omitted, to prevent parasitic contact compliance and efforts for contact handling. A discrete element formulation is used for a computationally efficient simulation of transverse loads. This prevents a reduction of the critical time step, which can result from transversal deformations. The difference between the two modelling approaches lies in the modelling of the out-of-plane behaviour. In the first approach, the behaviour in the out-of-plane direction is generated by characterising each component separately. The second approach assumes homogenous behaviour in the thickness direction. Both approaches rely on characterization data derived from component and cell tests. The cell models are validated against cell tests. Numerical results obtained with the two modelling approaches differ only slightly from the validation data.

Numerical modelling of thermal weakening of granite under dynamic loading

A numerical method to predict thermal weakening of granite rock under low- and moderate rate dynamic loading conditions is developed. Thermal weakening of granite under uniform heating, leading to degradation of the material stiffness and strength, is modelled with the continuum approach, using a damage-viscoplasticity model. The viscoplastic part is based on a special power law criterion for granite under compression and the rounded tensile cut-off surface, while the damage part is equipped with separate damage variables in tension and compression. The strain rate sensitivity of rock is accounted for by viscoplasticity. The global thermo-mechanical initial/boundary value problem is solved with an explicit (in time) staggered method with mass scaling applied to increase the critical time step. Rock heterogeneity is described as random clusters of linear tetrahedral finite elements assigned with the constituent mineral, here quartz, feldspar, and biotite, material properties. The temperature dependence of the thermal and elasticity properties of the minerals is included in the model up to 850 °C, i.e. well beyond the α-β transition of quartz. However, as the model aims to predict the thermal weakening, the temperature dependence of the rock strength is not used as an input. In the numerical examples, low-rate uniaxial tension and compression tests are first simulated on heated and intact rock samples to demonstrate the performance of the model. Finally, crushing of heat-treated and intact granite balls by diametral dynamic compression is numerically replicated with a fair accuracy.

Diurnal, physics-based strategy for computationally efficient capacity-expansion optimizations for solar-dominated grids

Modeling energy storage for a renewables-driven grid using every hour of the year gives more insight and higher accuracy, but can be computationally demanding. In this study, we propose a novel and straightforward Critical Time Step technique, independent of the load and generation profiles, to reduce computational requirements with little loss in accuracy for a solar-dominated system. Using the diurnal cycle, specifically the critical times of an hour after sunrise and an hour before sunset, results in an excellent tradeoff of accuracy and computational complexity compared with fixed-time-step approaches. The accuracy and value of the technique are evaluated by comparing results for three weather years. The technique systematically underestimates the capacity expansion needed but differentiates the three weather years with results correlating well with the hourly simulations. Overall, the results show a high accuracy for the Critical Time Step technique in predicting the power expansion of the resources and the energy rating expansion of the storage system for a grid with more than 35% share of solar in the total operational power. The highest error occurred for storage power buildout but did not exceed 10% relative to the 1-hr-resolution simulation for the studied case.

Explicit asynchronous time scheme with local push-forward stepping for discontinuous elastic wave propagation: One-dimensional heterogeneous cases and Hopkinson bar experiment

This is a presentation of robust and accurate explicit time-stepping strategy for finite element modeling of elastic discontinuous wave propagation in strongly heterogeneous, multi-material and graded one-dimensional media. One of the major issues in FEM modeling is the existence of spurious numerical stress oscillations close to theoretical wave fronts due to temporal-spatial dispersion behavior of FE discretization. The numerical strategy presented for modeling of 1D discontinuous elastic waves is based on (a) pushforward-pullback local stepping — ensuring the elimination of dispersion due to different critical time step sizes of finite elements, (b) domain decomposition via localized Lagrange multipliers — to satisfy coupling kinematics and dynamic equations , (c) asynchronous time scheme — ensuring the correct information transfer of quantities for the case of integer ratios of time step size for all domain pairs. Dispersion behaviors, existence of spurious stress oscillations, and sensitivity of the dispersion to time step size are then suppressed. The proposed method is numerically tested with regard to the rectangular step pulse elastic propagation problem considering in-space varying Young’s modulus. To prove robustness and accuracy, a comparison with results from commercial software, an analytical solution, and experimental data from partial assembly of a split Hopkinson pressure bar (SHPB) setup is provided.

Critical time-step size analysis and mass scaling by ghost-penalty for immersogeometric explicit dynamics

In this article, we study the effect of small-cut elements on the critical time-step size in an immersogeometric explicit dynamics context. We analyze different formulations for second-order (membrane) and fourth-order (shell-type) equations, and derive scaling relations between the critical time-step size and the cut-element size for various types of cuts. In particular, we focus on different approaches for the weak imposition of Dirichlet conditions: by penalty enforcement and with Nitsche’s method. The conventional stability requirement for Nitsche’s method necessitates either a cut-size dependent penalty parameter, or an additional ghost-penalty stabilization term. Our findings show that both techniques suffer from cut-size dependent critical time-step sizes, but the addition of a ghost-penalty term to the mass matrix serves to mitigate this issue. We confirm that this form of ‘mass-scaling’ does not adversely affect error and convergence characteristics for a transient membrane example, and has the potential to increase the critical time-step size by orders of magnitude. Finally, for a prototypical simulation of a Kirchhoff–Love shell, our stabilized Nitsche formulation reduces the solution error by well over an order of magnitude compared to a penalty formulation at equal time-step size.

A transient global-local generalized FEM for parabolic and hyperbolic PDEs with multi-space/time scales

This paper presents a novel Generalized Finite Element Method with global-local enrichment (GFEMgl) to solve time-dependent parabolic and hyperbolic problems with subscale features in both space and time. Compared with currently available GFEMgl, the proposed method requires a solution of local problems to solve transient PDEs, possibly with a different time integrator from the global problem or other local problems. The fine-scale solution can be easily retrieved. The proposed method is tested for the solution of the heat, advection, and advection-diffusion equations whose accuracy, stability, scalability, and convergence are thoroughly analyzed. Compared to direct analysis, numerical results show that the proposed method has the following properties 1) the accuracy closely matches direct analysis result with a fine mesh; 2) critical time step size is loosened; 3) optimal convergence rate is achieved. Moreover, the proposed method allows local problems to be turned off if their solutions are not helpful for the current global time step, which can further improve its efficiency.