Dynamic Field Model Research Articles

Dynamic cluster field modeling of collective chemotaxis

Collective migration of eukaryotic cells is often guided by chemotaxis, and is critical in several biological processes, such as cancer metastasis, wound healing, and embryogenesis. Understanding collective chemotaxis has challenged experimental, theoretical and computational scientists because cells can sense very small chemoattractant gradients that are tightly controlled by cell-cell interactions and the regulation of the chemoattractant distribution by the cells. Computational models of collective cell migration that offer a high-fidelity resolution of the cell motion and chemoattractant dynamics in the extracellular space have been limited to a small number of cells. Here, we present Dynamic cluster field modeling (DCF), a novel computational method that enables simulations of collective chemotaxis of cellular systems with O(1000)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(1000)$$\\end{document} cells and high-resolution transport dynamics of the chemoattractant in the time-evolving extracellular space. We illustrate the efficiency and predictive capabilities of our approach by comparing our numerical simulations with experiments in multiple scenarios that involve chemoattractant secretion and uptake by the migrating cells, cell migration in confined spaces, regulation of the attractant distribution by cell motion, and interactions of the chemoattractant with an enzyme. The proposed algorithm opens new opportunities to address outstanding problems that involve collective cell migration in the central nervous system, immune response and cancer metastasis.

The dynamic response of solid oxide fuel cell fueled by syngas during the operating condition variations

This work investigates the dynamic characteristics of solid oxide fuel cell (SOFC) under variation operating conditions and complements the dynamic response characteristics on SOFC operating end. The dynamic characteristics is significant to dynamic operation and performance control. The classical appropriate transfer functions are proposed to capture the practical changing process of operating conditions for the first time. The dynamic multi-physical field model is developed as a mapping function to obtain the dynamic characteristics. The SOFC will be stabilized again within 100 s after the flow rate variations, which means that SOFC can be in another stable operation. The current density obviously goes through an overshoot phase and then gradually reaches stability caused by cathode flow rate variation. While, the change of anode flow does not cause overshoot. It is also found that the “fuel starvation” phenomenon also does not exist during flow rate variations, which means the “fuel starvation” hardly needs to be considered under non-extreme changes in operating conditions. The step function is used to simulate the change process of syngas composition. The increase in current density can be divided into two stages, which last about 100 s totally. The variation amplitude of current density due to the fuel composition variation is relatively small, which means the change in fuel composition doesn't cause a very dramatic response. The cell length has relative significance influence on the dynamic characteristic. This dynamic model could also be used to simulate the dynamic response of SOFC with different geometric structures, especially those with large changes in length and direction. This work is expected to uncover the dynamic response characteristics of SOFC under the variations of operating parameters, and to provide data support and theoretical guidance for SOFC under variable operating conditions and performance control.

Flow Field Simulation Tool of Solid Propulsion Control Systems for Missile Applications

In order to achieve hit-to-kill miss distances against manoeuvrable ballistic missile targets, future generation missiles calls for fast response non-conventional propulsion control systems during terminal homing period in high-endo to exo-atmosphere regions augmenting aero control. Based on advances in thrust modulation controlling throat area in solid propulsion pintle motor, control systems are developed for missile Divert and Attitude Control Systems (DACS) applications using multiple nozzles in pitch/yaw directions connected to common gas generator. Control thrust vector for lateral and attitude corrections is generated with proportional distribution of total thrust in pitch/yaw directions through real time thrust modulation of pintle nozzles
 Towards development of propulsion control systems based on solid propulsion, prototype systems: Thrust Controlled Motor (TCM) and Divert Thruster Flight Propulsion Control System (DTFPCS) are developed to demonstrate thrust modulation/ control thrust vector generation in solid propulsion and finalize unified controller for various missile control applications. Design & development process of prototype systems calls for flow field simulation tool for parametric design analysis/optimization and generate performance maps/control parameters based on detailed flow distributions. Present paper presents flow field simulation tools with Steady Flow Field (SFF) model and Dynamic Flow Field (DFF) model integrating dynamics of pintle movement with steady flow solver. Parametric analysis of geometrical profiles of pintle/throat regions and effect of pintle speed on system response are presented for system design optimization. Performance maps and controller maps of prototype systems are generated based on flow field distributions for nozzle thrust modulation. Simulated results are compared with open loop hot tests and unified controller parameters are updated for control thrust vector generation. Finally flow field distributions of DTFPCS operating modes: neutral-thrust and lateral-thrust are presented demonstrating the utility of simulation tool in design analysis and system development. Present simulation tool is critical in system design optimization and unified controller development for various flight control applications with minimum number of developmental tests. Further work of integrating propellant grain geometry variation during motor hot tests, with flow field model is in progress to predict control system performance for various missile control applications.

Capturing Luminous Flux Entering Human Eyes with a Camera, Part 2: A Field Verification Experiment

ABSTRACT A space occupant’s real-time exposure to light reaching his/her eyes has been a concern in lighting research and design practice. Following the parallel fundamental research in Part 1, Part 2 of this study explored a way to use a Canon 550D fitted with the Sigma 4.5 mm F2.8 circular fisheye lens to estimate the luminous flux arriving at the entrance pupil from the visible luminous environment. By taking two 180° HDR images at the same camera position but in different aiming directions and merging them into a single luminance map covering the observer’s entire binocular visual field (214° horizontally and 125° vertically), the monocular and binocular luminance map can be generated with the aid of the dynamic visual field model proposed in Part 1. In a field verification experiment, the luminous flux passing through the entrance pupil in the preset scenes and gaze angles was estimated. Individual luminous contribution from an object or a region of interest within the occupant’s dynamic binocular visual field or subfields was also calculated. This study expanded the capacity of HDR photography in lighting measurement to capture binocular luminous flux entering human eyes as input for optimizing human-light interaction in various luminous environments.

A Holistic Approach for Role Inference and Action Anticipation in Human Teams

The ability to anticipate human actions is critical to many cyber-physical systems, such as robots and autonomous vehicles. Computer vision and sensing algorithms to date have focused on extracting and predicting visual features that are explicit in the scene, such as color, appearance, actions, positions, and velocities, using video and physical measurements, such as object depth and motion. Human actions, however, are intrinsically influenced and motivated by many implicit factors such as context, human roles and interactions, past experience, and inner goals or intentions. For example, in a sport team, the team strategy, player role, and dynamic circumstances driven by the behavior of the opponents, all influence the actions of each player. This article proposes a holistic framework for incorporating visual features, as well as hidden information, such as social roles, and domain knowledge. The approach, relying on a novel dynamic Markov random field (DMRF) model, infers the instantaneous team strategy and, subsequently, the players’ roles that are temporally evolving throughout the game. The results from the DMRF inference stage are then integrated with instantaneous visual features, such as individual actions and position, in order to perform holistic action anticipation using a multi-layer perceptron (MLP). The approach is demonstrated on the team sport of volleyball, by first training the DMRF and MLP offline with past videos, and, then, by applying them to new volleyball videos online. These results show that the method is able to infer the players’ roles with an average accuracy of 86.99%, and anticipate future actions over a sequence of up to 46 frames with an average accuracy of 80.50%. Additionally, the method predicts the onset and duration of each action achieving a mean relative error of 14.57% and 15.67%, respectively.

A dynamic phase field model for predicting rock fracture diversity under impact loading

Predicting dynamic rock fracture is of great significant in rock engineering. The phase field model (PFM) seems to a suitable numerical tool for simulating dynamic rock fracture. However, dynamic fracture diversity cannot be well reproduced by the existing PFMs. Only tension-induced dynamic fracture can be modeled while the dynamic compressive-shear and mixed-mode fractures are limited. Therefore, this study aims to propose a new phase field model that involves all the commonly seen dynamic fracture mechanisms to reflect dynamic rock diversity under impact loading. A new driving force is used and the hybrid phase field framework is adopted. Our PFM is implemented in the framework of finite element method and a staggered scheme is used to solve the coupled governing equations. Previous numerical and experimental studies are used to initially verify the proposed method. A dynamic mode-II test and a dynamic Brazilian disc test are simulated to show the applicability and feasibility of the proposed PFM and also to investigate the dynamic rock fracture diversity under impact loading. The numerical results indicate that the proposed PFM inherits the advantages of the conventional PFMs. Crack initiation and propagation are automatically characterized by the evolution equations of phase field. The parameters used in our model are more suitable for rocks. The difference between tensile-shear fracture and compressive-shear fracture is well distinguished. Different fracture patterns can be predicted by using different coefficient combinations in the proposed dynamic PFM.

Construction of dynamic temperature field model of ball screw based on superposition of positive and negative temperature fields

In order to study the influence of internal heat source of ball screw feed transmission system on the thermal error of ball screw, the theoretical modeling and experimental research are carried out on the temperature field and temperature rise characteristics of ball screw in this paper. Firstly, on the basis of analyzing heat generation mechanism and heat transfer of ball screw, the positive temperature field of ideal heat transfer of heat source and the negative temperature field caused by heat transfer are derived. In addition, the mathematical model of dynamic temperature field of ball screw feed system about time and position is established according to the superposition principle of temperature field, and the temperature field and thermal error of x-axis ball screw in horizontal machining center are measured. Secondly based on the experimental temperature field data, a complete finite element model of the temperature field of the ball screw feeding system is constructed, and temperature at any point of the system and the undetermined coefficient of the numerical model of temperature field under this working condition are determined by the finite element model. Finally, the thermal error distribution of the ball screw is calculated by the numerical model, the theoretical and experimental error distributions are compared, and the correctness of the theoretical model of the temperature field under this working condition is verified.

Testing predictions of a neural process model of visual attention in infancy across competitive and non-competitive contexts.

A key question in early development is how changes in neural systems give rise to changes in infants' behavior. We examine this question by testing predictions of a dynamic field (DF) model of infant spatial attention. We tested 5‐, 7‐, and 10‐month‐old infants in the Infant Orienting With Attention (IOWA) task containing the original non‐competitive cue conditions (when a central stimulus disappeared before a cue onset) and new competitive cue conditions (when a central stimulus remained visible throughout the trial). This allowed testing of five model predictions: (1) that orienting accuracy would be higher and (2) reaction times would be slower for all competitive conditions; (3) that all infants would be slower to orient in the competitive conditions, though (4) older infants would show the strongest competition costs; and (5) that reaction times would be particularly slow for un‐cued competitive conditions. Four of these five predictions were supported, and the remaining prediction was supported in part. We next examined fits of the model to the expanded task. New simulation results reveal close fits to the present findings after parameter modification. Critically, developmental parameters of the model were not altered, providing support for the DF model's account of neuro‐developmental change.

Dynamical effective field model for interacting ferrofluids: II. The proper relaxation time and effects of dynamic correlations

The recently proposed dynamical effective field model (DEFM) is quantitatively accurate for ferrofluid dynamics. It is derived in paper I within the framework of dynamical density functional theory (DDFT) along with a phenomenological description of nonadiabatic effects. However, it remains to clarify how the characteristic rotational relaxation time of a dressed particle, denoted by τ r , is quantitatively related to that of a bare particle, denoted by . By building macro-micro connections via two different routes, I reveal that under some gentle assumptions τ r can be identified with the mean time characterizing long-time rotational self-diffusion. I further introduce two simple but useful integrated correlation factors, describing the effects of quasi-static (adiabatic) and dynamic (nonadiabatic) inter-particle correlations, respectively. In terms of both the dynamic magnetic susceptibility is expressed in an illuminating and elegant form. Remarkably, it shows that the macro-micro connection is established via two successive steps: a dynamical coarse-graining with nonadiabatic effects accounted for by the dynamic factor, followed by equilibrium ensemble averaging captured by the static factor. By analyzing data from Brownian dynamics simulations on monodisperse interacting ferrofluids, I find is, somehow unexpectedly, insensitive to changes of particle volume fraction. A physical picture is proposed to explain it. Furthermore, an empirical formula is proposed to characterize the dependence of on dipole-dipole interaction strength. The DEFM supplemented with this formula leads to parameter-free predictions in good agreement with results from Brownian dynamics simulations. The theoretical developments presented in this paper may have important consequences to studies of ferrofluid dynamics in particular and other systems modeled by DDFTs in general.

Dynamical effective field model for interacting ferrofluids: I. Derivations for homogeneous, inhomogeneous, and polydisperse cases

Quite recently I have proposed a nonperturbative dynamical effective field model (DEFM) to quantitatively describe the dynamics of interacting ferrofluids. Its predictions compare very well with the results from Brownian dynamics simulations. In this paper I put the DEFM on firm theoretical ground by deriving it within the framework of dynamical density functional theory, taking into account nonadiabatic effects. The DEFM is generalized to inhomogeneous finite-size samples for which the macroscopic and mesoscopic scale separation is nontrivial due to the presence of long-range dipole–dipole interactions. The demagnetizing field naturally emerges from microscopic considerations and is consistently accounted for. The resulting mesoscopic dynamics only involves macroscopically local quantities such as local magnetization and Maxwell field. Nevertheless, the local demagnetizing field essentially couples to magnetization at distant macroscopic locations. Thus, a two-scale parallel algorithm, involving information transfer between different macroscopic locations, can be applied to fully solve the dynamics in an inhomogeneous sample. I also derive the DEFM for polydisperse ferrofluids, in which different species can be strongly coupled to each other dynamically. I discuss the underlying assumptions in obtaining a thermodynamically consistent polydisperse magnetization relaxation equation, which is of the same generic form as that for monodisperse ferrofluids. The theoretical advances presented in this paper are important for both qualitative understanding and quantitative modeling of the dynamics of ferrofluids and other dipolar systems.

Sequential method for fast neural population activity reconstruction in the cortex from incomplete noisy measurements

During recent years there has been a growing interest in stochastic dynamic neural fields employed for modeling and predictions in biomedical and technical systems. In this paper, given some incomplete noisy data available from sensors, we propose and explore a state estimation method for fast restorations of membrane potential in the cortex based on such measurements and the Amari equation used for simulations of neural population activity in a stochastic setting. Our novel technique relies upon a Galerkin-type spectral approximation utilized within the conventional state-space approach. Translating a stochastic system into its state-space form creates a straightforward and fruitful way to the data-driven parameter estimation, filtering, prediction and smoothing. The present study is particularly focused on establishing a nonlinear stochastic Galerkin-spectral-approximation-induced system of large size, which is further estimated by the traditional extended Kalman filter (EKF). The efficiency of calculations is the main purpose of our research. That is why the fast filtering solution devised is based on processing the incoming data incrementally, that is, by processing measurements one at a time, rather than handling them as a unified high-dimensional vector. Such sequential filters suit well for dealing with large data sets as well as with real-time on-line computations. Also, their derivation and substantiation is of great interest in the context of neural network training because of large stochastic systems arisen there. In comparison to the batch filtering, our novel algorithm reduces the computational cost of membrane potential reconstructions in terms of the amount of grid nodes N accepted in the underlying spacial discretization, significantly. Apart from its computation efficiency, this sequential method is more robust to round-off errors committed within a computer-based finite precision arithmetics than the classical EKF because of the (N × N)-matrix inversion elimination from such membrane potential calculations. The superior performance of our technique is examined and confirmed in comparison to the batch one on two known scenarios in the dynamic neural field modeling.

Time-domain dynamic electric field distribution of low-oscillation sculptured structure dispersive mirrors

The dispersion oscillation and damage threshold of a dispersive mirror (DM) are the key influencing factors for high-quality pulses and outputting high power density. Here, we studied the femtosecond damage behavior of low-oscillation DMs(LODM) with SiO2 sculptured layer. The monochromatic standing wave field(SWF) distribution widely used in ultrafast laser damage analysis is sufficient in Bragg mirror. In more complex systems, the actual electric field intensity(typically smaller) will dynamically change due to the broad bandwidth and the time difference of pulse propagation. Time-domain dynamic electric field intensity displays that the spatio-temporal evolution of the pulse within the stacks averages the electric field intensity, especially the electric field in the deeper layers. The dynamic effect of few-cycle pulses or attosecond pulses damage will be more meaningful. The time-domain dynamic electric field model can elucidate the ultrafast laser damage behavior. Introducing the time-domain dynamic electric model into the merit function can help us design optical coatings with high damage thresholds.

Theoretical Model and Experimental Study of Dynamic Hot Rolling

At present, the commonly used hot rolling model is only applicable to the static rolling process. However, to study the dynamic rolling process, a dynamic rolling model with roll vertical movement velocity parameters is required. In this study, the influence of the vertical movement velocity of the rolls on the rolling process is considered, and a dynamic rolling process model is proposed when the roll gap is reduced during the hot rolling dynamic rolling process. Mathematically, the model is based on the upper limit method. The model considers the influence of the dynamic rolling process on the length of the deformation zone, establishes a dynamic velocity field model and an average deformation rate model, and then solves the total power of the rolling process. Finally, the dynamic rolling force equation is given. Compared with the experimental results, the dynamic rolling model in this paper has high accuracy with an average error of 4%. In addition, the influence of roll vertical velocity on rolling parameters is discussed, which provides a basis for the study of the dynamic rolling process.

Vortex core reversal by elastic waves in ferromagnetic materials

Magnetic vortex has attracted increasing attention due to its potential application in magnetic logic and memory devices. With the development of straintronics, acoustic waves could remotely control the topologically nontrivial magnetic bits such as magnetic vortices. Understanding the dynamic response of magnetic vortex subjected to elastic waves is crucial for the future logic and memory devices. In the present work, a dynamic phase field model is developed to investigate the dynamic magnetoelastic coupling behavior between elastic waves and magnetic vortices. Phase field simulations show that the magnetic vortex core can flip over under the elastic waves with gigahertz frequency. Compared with the results of conventional phase field model without the elastodynamic term, it is found that the strain gradient caused by elastic wave plays an important role in the reversal of vortex core. When the strain gradient in the ferromagnetic material reaches a certain value, the vortex core is likely to flip over. The elastic waves with larger magnitudes and frequencies have the larger strain gradient, which are more easily to cause the reversal of vortex core. In addition to the strain gradient, magnetic damping constant also has great influence on the vortex core reversal by changing the precession process of magnetization. Larger magnetic damping constant makes the smaller precession of magnetization, which retards the reversal of vortex core. The results of this study suggest a mechanical way for the remote control of magnetic vortices by elastic waves.

Temperature Effect on Stability of Clamped-Clamped Composite Annular Plate with Damages.

The paper presents the response of a three-layered annular plate with different damaged laminate facings to the action of the static or dynamic temperature field model. Various damages of laminate, composite facings change the plate structure reaction under the temperature fields. Obtained results indicate practical meaning of analyses in failure diagnostic process. The thermal sensitivity of two kinds of plate structures, undamaged and damaged, offers both new practical and scientific possibilities in evaluation of the plate behavior. The relations between macro-damage, i.e., the buckling of the plate structure and micro-damages of plate layers subjected to temperature gradient, are shown. The numerical solution is proposed as the most effective in examinations of the various transversally symmetrical and asymmetrical plate structures with a different rate of damages. The graphical distribution of changes in values of static and dynamic critical loads illustrate the process of structural damaging during its exploitation. They have practical importance in the evaluation of the structure capacity. The knowledge of the effect of laminate degradation process on plate buckling phenomenon located in thermal environment complements previous investigations and designates complex, multi-parameter problems as having scientifically new elements.

Phase field study on the performance of artificial synapse device based on the motion of domain wall in ferroelectric thin films

Artificial neural networks have gained intensive attention in recent years because of their potential in effectively reducing energy consumption and improving computation performance. Ferroelectric materials are considered to be promising candidates for artificial synapses because of their multiple and nonvolatile polarization states under external stimuli. Despite artificial ferroelectric synapses with multilevel states, long retention and fast switching speed have been reported, and some key fundamental issues, e.g., the influence of domain wall configuration and evolution on the performance of synapse behaviors, also remain unclear. In this work, we study the performance of artificial synapses based on the motion of 180° ferroelectric domain walls of stripe domain and cylinder domain in ferroelectric thin films via a dynamical phase field model. The results demonstrate that artificial synapses based on the stripe domain exhibit high linearity and symmetry in weight update under a weak electric field, compared with the cylinder domain. Based on such artificial synapses, the accuracy of an artificial neural network for the Modified National Institute of Standards and Technology handwritten digit recognition is over 92%. This work provides a domain-wall-based strategy to improve the weight updating linearity and symmetry of artificial synapse devices and the recognition accuracy of artificial neural networks.

Optimum Integration Procedure for Connectionist and Dynamic Field Equations.

Connectionist and dynamic field models consist of a set of coupled first-order differential equations describing the evolution in time of different units. We compare three numerical methods for the integration of these equations: the Euler method, and two methods we have developed and present here: a modified version of the fourth-order Runge Kutta method, and one semi-analytical method. We apply them to solve a well-known nonlinear connectionist model of retrieval in single-digit multiplication, and show that, in many regimes, the semi-analytical and modified Runge Kutta methods outperform the Euler method, in some regimes by more than three orders of magnitude. Given the outstanding difference in execution time of the methods, and that the EM is widely used, we conclude that the researchers in the field can greatly benefit from our analysis and developed methods.

Deconvolving Contributions to Decoherence in Molecular Electron Spin Qubits: A Dynamic Ligand Field Approach.

In the past decade, transition metal complexes have gained momentum as electron spin-based quantum bit (qubit) candidates due to their synthetic tunability and long achievable coherence times. The decoherence of magnetic quantum states imposes a limit on the use of these qubits for quantum information technologies, such as quantum computing, sensing, and communication. With rapid recent development in the field of molecular quantum information science, a variety of chemical design principles for prolonging coherence in molecular transition metal qubits have been proposed. Here the spin-spin, motional, and spin-phonon regimes of decoherence are delineated, outlining design principles for each. It is shown how dynamic ligand field models can provide insights into the intramolecular vibrational contributions in the spin-phonon decoherence regime. This minireview aims to inform the development of molecular quantum technologies tailored for different environments and conditions.

The Non-mechanistic Option: Defending Dynamical Explanations

This article demonstrates that non-mechanistic, dynamical explanations are a viable approach to explanation in the special sciences. The claim that dynamical models can be explanatory without reference to mechanisms has previously been met with three lines of criticism from mechanists: the causal relevance concern, the genuine laws concern, and the charge of predictivism. I argue, however, that these mechanist criticisms fail to defeat non-mechanistic, dynamical explanation. Using the examples of Haken et al.’s ([1985]) model of bimanual coordination, and Thelen et al.’s ([2001]) dynamical field model of infant perseverative reaching, I show how each mechanist criticism fails once the standards of Woodward’s ([2003]) interventionist framework are applied to dynamical models. An even-handed application of Woodwardian interventionism reveals that dynamical models are capable of producing genuine explanations without appealing to underlying mechanistic details. 1. Introduction2. Interventionism and Mechanistic Explanation 2.1. Causal relevance and ideal interventions2.2. Invariance2.3. Explanation3. Covering-Laws and Dynamical Explanation 3.1. Dynamical models3.2. Covering-law explanation3.3. Prediction4. Causal Relevance 4.1. The causal relevance concern4.2. Intervening on dynamical models4.3. Test case I: The Haken–Kelso–Bunz model4.4. Test case II: Dynamical field model5. Genuine Laws 5.1. The genuine laws concern5.2. Using invariance in place of laws5.3. Test case I: The Haken–Kelso–Bunz model5.4. Test case II: Dynamical field model6. Prediction 6.1. Predictivism6.2. Crude and invariant prediction7. Interventionist Criticism of the Haken–Kelso–Bunz Model8. Dynamical Explanation8. Conclusion

A dynamic phase field model with no attenuation of wave speed for rapid fracture instability in hyperelastic materials

Numerical experiments on phase field modeling of the fracture in pre-strained hyperelastic materials reveal that the classical mechanical-based dynamic phase field model is ineffective in the framework of non-linear deformation. The aspiration to gain insight into rapid fracture instability motivated us to develop a novel dynamic phase field model characterized by wave velocity invariance, enabling crack propagation at a velocity approach to the asymptotic limit. Given that the numerical treatment of rapid fractures involves extremely high spatiotemporal resolution, robust explicit dynamics and a tried-and-tested multi-level hybrid adaptive mesh algorithm are invoked. More crucially, an original adaptive distorted mesh removal scheme (ADMR) was developed to cope with the intractable finite element mesh distortion problem in large deformation fractures. The detailed numerical implementation for entire procedures is outlined, and its reliability is verified by two quasi-static fracture benchmarks. Utilizing the proposed model and innovative algorithms, the arresting ultrahigh-speed crack oscillation and tip-splitting instabilities captured in the fracture experiments of brittle gels were successfully reproduced. The critical crack velocity at the onset of the instability is also identified, close to the experimental measurements.