Corrective Forces Research Articles

GRACE-FO attitude determination: Star camera installation matrix calibration and incremental quaternion integrator

The satellite attitude error is transmitted to the gravity field through KBR antenna phase center correction and non-conservative force conversion, which affects the accuracy of gravity field inversion. Therefore, obtaining high-precision satellite attitude is one of the important research contents of gravity field inversion. Each satellite of GRACE-FO is equipped with three SCs and an IMU for satellite attitude measurement, and fusion of these two types of data can obtain high-precision satellite attitude. In this paper, we use kalman filter to fuse the original GRACE-FO 1A observation data, and take into account the calibration of the installation matrix of the SCs during the fusion process. Meanwhile, a quaternion integrator that takes angle increments as inputs is derived from the equivalent rotation vector differential equation to address the computational inefficiency of first-order quaternion integrator, the accuracy and computational efficiency are verified using measured data. The experimental results show that after the calibration of the installation matrix of the SCs, the system bias of 20′′ caused by switching from three SCs observations to two SCs observations can be eliminated. The standard deviation of the attitude difference between the obtained attitude in this paper and the attitude calculated by JPL is at most 6′′ and the proposed incremental quaternion integrator has the same accuracy as the first-order quaternion integrator but with computational efficiency improved by at least 35%.

Finite size effects and optimization of the calculation of the surface tension in surfactant mixtures at liquid/vapour interfaces

The surface tension of monolayers with mixtures of anionic and nonionic surfactant at the liquid/vapour interface is studied. Previous works have observed that calculations of the surface tension of simple fluids show artificial oscillations for small interfacial areas, indicating that the surface tension data fluctuate due to the finite size effects and periodic boundary conditions. In the case of simulations of monolayers composed of surfactant mixtures, the surface tension not only oscillates for small areas but can also give non-physical data, such as negative values. Analysis of the monolayers with different surfactant mixtures, ionic (DTAB, CTAB, SDS) and nonionic (SB3-12), was done for density profiles, parameters of order and pair correlation functions for small and large box areas and all of them present similar behaviour. The fluctuations and the non-physical values of the surface tension are corrected when boxes with large interfacial areas are considered. The results indicate that in order to obtain reliable values of the surface tension, in computer simulations, it is important to choose not only the correct force field but also the appropriate size of the simulation box.

Effect of Strouhal number on propulsion of tandem flapping foils

Motivated by the energy efficient locomotion by flock of birds and school of fish to develop biomimetic systems, the current study studies the propulsive performance of the tandem configuration of two-foil system. The effects of the kinematic motion parameters such as heave and pitch amplitudes and flapping frequency on the propulsive performance of the tandem two-foil system is yet to be explored comprehensively due to the complex nature of the wake-foil interaction. Here, we numerically investigate these effects by utilizing a finite element based moving mesh framework to model the flapping foil motion subjected to fluid flow. The performance of the upstream foil is found to be similar to that of an isolated single foil. However, for the downstream foil, a beating phenomenon is observed in the thrust response when the flapping frequency is different for both the foils. This feature can help tune propulsive performance of the foil system. Furthermore, through a control volume analysis, we show that sole reliance on the time-averaged streamwise velocity may not yield correct trends for the mean thrust forces, which has been typically followed in the literature. Variation in the time-averaged pressure in the wake is significant in predicting correct thrust forces.

Stabilizing nodal integration in dynamic smoothed particle finite element method: A simple and efficient algorithm

The Smoothed Particle Finite Element Method (SPFEM) is a powerful numerical tool for the modelling of large deformation problems in the field of geomechanics. However, the standard SPFEM with nodal integration suffers from the rank deficiency problem due to under-integration. In this study, an hourglass-control based stabilization scheme is presented to overcome the rank deficiency in standard SPFEM with nodal integration. By introducing the hourglass-control based correction forces, the spurious low-energy models in standard SPFEM with nodal integration can be effectively suppressed. The presented stabilization scheme is simple and computationally efficient compared to the existing strain gradient-based stabilization approaches in SPFEM. The proposed hourglass-control based stabilized SPFEM is firstly validated by means of three benchmark examples and the results show that the stabilization scheme based on the hourglass-control can eliminate the rank deficiency instability in standard SPFEM and free of volumetric locking. Finally, two numerical examples related to the cohesive soil flow and the progressive failure of sensitive clay slopes are studied to highlight the robustness of the proposed stabilized SPFEM in solving large deformation geotechnical problems.

PID Optimization Tuning Using Lagrangian Mechanics

Creating a simulation of a system enables the tuning of control systems without the need for a physical system. In this paper, we employ Lagrangian Mechanics to derive a set of equations to simulate an inverted pendulum on a cart. The system consists of a freely-rotating rod attached to a cart, with the rod’s balance achieved through applying the correct forces to the cart. We manually tune the proportional, integral, and derivative gain coefficients of a Proportional Integral Derivative controller (PID) to balance a rod. To further improve PID performance, we can optimize an objective function to find better gain coefficients.

Obstacle Avoidance Model of Two-Wheeled Vehicles for Nonrigid Obstacles Based on Social Forces

To describe the microtrajectory change behavior of two-wheeled vehicles when encountering nonrigid obstacles, this paper proposes an obstacle avoidance model of two-wheeled vehicles for nonrigid obstacles based on the improved social force method. In the model, the obstacle avoidance characteristics of two-wheeled vehicles for nonrigid obstacles and the safety operation requirements of two-wheeled vehicles are considered. The calculation methods of the minimum boundary distance of avoidance and the boundary avoidance force range edge function are proposed based on the ultimate turning angle and boundary avoidance characteristics, respectively. The speed control and direction control behavior of the rider to the vehicle is abstracted as the form of social force, and the two-wheeled vehicle/rider individual is subjected to the avoidance force and correction force perpendicular to the direction of the longitudinal axis of the vehicle body. The centrifugal trajectory obstacle avoidance motion obeying the Newtonian mechanics is studied based on the characteristics of the vehicle body turning angle change. Different traffic environments are constructed by MATLAB software and simulated according to the model running logic, and sensitivity analysis is performed based on the actual data collected in the survey. The simulation results show that, under the same conditions, compared with μi = 50N, when μi = 200N, the X-axis displacement of the simulated vehicle in the avoidance process is shortened by 26.31%, and the body deflection angle increases by 50.75% at the end moment of the avoidance process. Compared with electric motorcycles, the displacement of the X-axis in the avoidance process of the electric bicycle is shortened by 16.92%, and the body deflection angle at the end moment of the avoidance process is increased by 13.68%. The simulation results show that the model can well describe the trajectory change behavior characteristics of two-wheeled vehicles/riders encountering nonrigid obstacles in real situations, and the obstacle avoidance model of two-wheeled vehicles proposed in this paper is considered to be traffic justified.

Wonder of Wonders: How Nonstandard Lives Help Us to Remain Human.

Wonder is a gateway, not a machine. It is not something that can be functionalized for the purposes of education, moral uplift, or humanizing medicine. The things and experiences that evoke wonder today have their own history. Inhabitants of the Western developed world, for instance, have been taught to wonder at the power of science to control nature and at the ingenuity of the scientists and technicians who have invented the techniques of science and technology. This article examines what wonder tells us about our grasp of the human, and also why the language of wonder sometimes jars within the constraints of modern, liberal, scientifically inclined public discourse. The author suggests that privileging wonder at the appearance of the nonstandard human body is important for the lives of disabled people and also as a corrective force to ensure that the form of wonder that dominates the modern developed West-wonder at our own powers of scientific description and medical remaking of the human-not become inhumane.

An exploration into the orthopedic effect and patterns of the correction based on the three-point force system in patients with strephenopodia

According to the orthopedic principle based on the three-point force system, this study aimed to reveal the orthopedic effect and patterns of the orthopedic based on this system in 3 patients with strephenopodia through the weight-bearing stance phase experiment and numerical simulation analysis of ankle-foot orthoses (AFOs). After the correction with AFOs, Under the pressure tolerance of human skin, different three correction forces were applied to the medial heel, above the medial ankle, below the lateral ankle and the first metatarsal bone,the weight-bearing region of the affected foot gradually shifted to the normal region of the first metatarsal bone, the fifth metatarsal bone, and the heel. The weight-bearing capacity of the affected foot increased by 11.4 %. After the three-point force was adjusted for the AFO, the distribution of the plantar weight-bearing region became more uniform, and the supporting force of the affected foot increased significantly, the weight-bearing capacity of the affected foot increased to 15.19 %, which approached the theoretical value. When the three-point force of was over-adjusted to exceed the pain threshold, the weight-bearing capacity of the affected foot decreased by 2.79 % compared with the theoretical value. It was confirmed that the foot varus angle was attenuated significantly 8 weeks after the orthopedic with three-point force system. Further, the strephenopodia with 10° was corrected with the finite element method based on the three-point force principle, and orthopedic angle and the plantar weight-bearing region before and after the correction were also obtained, The visualization of orthopedic effects and the orthotic principle is clear contributes to clinicians provide important scientific reference for the formulation of scientific and accurate orthopedic regimens.

Finite Element Model of Scoliosis Brace with Increased Utility Characteristics

Orthoses are of critical importance in the field of medical biomechanics, particularly in the correction of spinal deformities. The objective of the current research was to improve the utility characteristics of the scoliosis brace without compromising its corrective capabilities. The orthotic shell of the Boston brace was used as the basis for this investigation. The finite element method (FEM) was used to evaluate the distribution of corrective forces through the device. The flow of force lines within the orthotic shell was determined by mapping the paths of maximum principal stresses. Areas of the device that had a negligible effect on overall stiffness were identified and material from these areas was eliminated. Minor modifications were then made to the redesigned shell to maintain its corrective stiffness. As a result of these changes, the weight of the braces was reduced without compromising its corrective stiffness. When subjected to corrective forces, the shell’s displacement patterns in the transverse plane showed minimal changes from the original model, confirming that its corrective capacity remained largely intact. This research presents an innovative methodology for orthotic design and demonstrates that structural optimization based on the mapping of maximum principal stress pathways can significantly improve device functionality. The approach outlined here holds promise for future advances in the design of various orthotic devices, thereby contributing to the advancement of the field.

Numerical study on the lifting operation of a gravity-type fish cage

The dynamic behaviours of the lifting operation of a gravity-type fish cage under calm sea conditions are investigated in this study using an extended position-based dynamics (XPBD) method to obtain the structural deformations of the aquaculture nets. The original XPBD is improved to accurately predict the tensions of the aquaculture nets by applying correction forces. The present XPBD is validated by comparing the experimental results of a flexible horizontal net. The time-step sensitivity is verified for the case of the lifting operation of the fish cage. Results show that the lifting force increases rapidly resulting from the weight of the sinkers at the bottom of the side net. The maximum tension of the net is located at the net ropes connected to the centre point of the bottom net. The structure of the bottom net should be enhanced for safety lifting operations.

Study of the corrective effect of different fixators on pectus excavatum during Nuss procedure

Background. A funnel chest is one of the most common chest deformities, which leads not only to cosmetic problems in adolescents, but also to cardiopulmonary complications. The main method of surgical correction is the Nuss procedure. The issues of the interaction between the fixator and the sternocostal joint depending on the choice of the plate length and the location of the tunnel for the fixator inside the chest to exit it on the opposite side remain undefined. Goal: to study the maximum relative deformities and displacements that occur in the chest model depending on the correction for pectus excavatum. Materials and methods. Four schemes for the correction of a funnel chest were modeled: 1) medial delivery of the fixator, the entry point is parasternal, using one retrosternal plate with transverse stabilizing bars (a short plate); 2) lateral passage of the fixator, the point of entry and exit from the chest is at the level of the anterior axillary line, using one retrosternal plate with transverse stabilizing bars; accordingly, the sternal plate is longer, ends at the level of the midaxillary line (a long plate); 3) a double plate with transverse bars connecting the plates with the help of screws (a short bridge-type fixator) with medial delivery; 4) a double plate with transverse bars connecting the plates with the help of screws (a long bridge-type fixator) with lateral delivery. The models were loaded with a distributed force of 100 N applied to the sternum. Results. When correcting pectus excavatum with a short plate, the cartilages of the fourth ribs turn out to be the most deformed — 3.3 %. In the cartilages of the ribs located above, deformities are in the range from 2.7 to 3.1 %. The use of a long plate decreases the relative deformities of the cartilage on almost all ribs. The scheme of correction using a short bridge-type fixator allows significantly reducing the deformities of all costal cartilages. The maximum is observed in the cartilage of the second and first ribs — 2.0 and 1.8 %, respectively. Replacing a short bridge-type fixator with a long one leads to the fact that the cartilages of the upper ribs remain deformed — 1.8 %, and a deformity gradually decreases to 1.0 % in the cartilages of the fourth ribs. The maximum movements in all schemes for pectus excavatum correction fall on the xiphoid process. The maximum displacement of 6.0 mm in the xiphoid process occurs when using a short plate. Replacing the plate with a long one decreases the displacement of the xiphoid process to 5.0 mm. When using a bridge-type fixator, the displacement of the xiphoid process is determined at the marks of 4 and 3 mm for a short and long fixator, respectively. Conclusions. All the investigated indicators testify to the advantages of a double bridge-type fixator. The medial passage of the fixator (short plates) has greater corrective forces on the anterior chest wall during elevation, which should be considered when choosing a correction technique. However, the lateral application of the fixator distributes the corrective effect by area, which can be important in preventing erosion of the tissues of the inner chest wall, the need for extended elevation of the depression in flat-concave forms of pectus excavatum, and the reduction of pain syndrome in the postoperative period.

Improvement on the axial force accuracy of the ANCF Euler-Bernoulli beam element with the second-order approximate function of the centerline and precise constraint equations

The conventional Euler-Bernoulli beam element of the absolute nodal coordinate formulation (ANCF) yields incorrect axial force due to the existence of the first-order approximate function of the centerline and over-constraint equations. This work aims to improve the axial force accuracy of the element by using the second-order centerline approximate function and precise constraint equations. The second-order function includes the quintic Hermite shape function and the vector of nodal coordinates containing zeroth to second-order derivatives of position. The elastic force vector of the element and its tangent stiffness were then formulated based on the decoupled elastic line method. The exact kinematic requirements of three types of constraint points were fully clarified. On this basis, the corresponding concise constraint equations, constraint Jacobian and Hessian matrices were rigorously derived in the context of ANCF, where G2 continuity was demonstrated to be satisfied at the node acted by an inclined concentrated force, the relative rotation constraint of the rigid joint was described using the dot product of the gradients on both sides of the joint, and the fixing gradient direction at the partially-clamped end was expressed by using the zero cross product of the end gradients in the initial and deformed configurations. Large deformation problems of typical beams under fixed and follower loads were solved using five types of elements, where the results of the ABAQUS B22 elements were taken as a reference for validation. It is found that when using the first- and second-order beam elements having over-constraint equations of the three types of constraint points, obvious axial force error occurs near the corresponding constraint points. In comparison, a few second-order elements containing the derived constraint equations output correct axial force consistent with ABAQUS. However, a large number of the first-order elements containing the correct constraint equations still yield inaccurate axial force in certain beam regions.

Numerical modelling of an orthopedic brace with increased functional characteristics for the treatment of idiopathic scoliosis.

Orthotic braces play a key role in the correction of spinal deformities. The effectiveness of these devices depends on the design and distribution of corrective forces transmitted through the corset shell. The present study aimed to reduce the weight of the orthosis and improve its functionality while maintaining its corrective function. The distribution of corrective forces transmitted by the orthosis was evaluated using the finite element method (FEM). Areas of the orthosis, which had minimal impact on the overall stiffness, were identified and material from these areas was removed. The modified orthosis shell was subjected to minor adjustments to maintain its corrective stiffness. With the modifications made, a 39% reduction in the weight of the orthosis was achieved, while maintaining its corrective stiffness. This indicates that the corrective function was largely preserved. The study provides a novel approach to orthosis design demonstrating that optimizing the structure using the distribution of maximum principal stress trajectories can significantly improve the functionality of the brace. The proposed method offers potential advances in the design of various types of orthoses, contributing to developments in the field.

A Cortical Mechanism Linking Saliency Detection and Motor Reactivity in Rhesus Monkeys.

Sudden and surprising sensory events trigger neural processes that swiftly adjust behavior. To study the phylogenesis and the mechanism of this phenomenon, we trained two male rhesus monkeys to keep a cursor inside a visual target by exerting force on an isometric joystick. We examined the effect of surprising auditory stimuli on exerted force, scalp electroencephalographic (EEG) activity, and local field potentials (LFPs) recorded from the dorsolateral prefrontal cortex. Auditory stimuli elicited (1) a biphasic modulation of isometric force, a transient decrease followed by a corrective tonic increase, and (2) EEG and LFP deflections dominated by two large negative-positive waves (N70 and P130). The EEG potential was symmetrical and maximal at the scalp vertex, highly reminiscent of the human "vertex potential." Electrocortical potentials and force were tightly coupled: the P130 amplitude predicted the magnitude of the corrective force increase, particularly in the LFPs recorded from deep rather than superficial cortical layers. These results disclose a phylogenetically preserved corticomotor mechanism supporting adaptive behavior in response to salient sensory events.Significance Statement Survival in the natural world depends on an animal's capacity to adapt ongoing behavior to abrupt unexpected events. To study the neural mechanisms underlying this capacity, we trained monkeys to apply constant force on a joystick while we recorded their brain activity from the scalp and the prefrontal cortex contralateral to the hand holding the joystick. Unexpected auditory stimuli elicited a biphasic force modulation: a transient reduction followed by a corrective adjustment. The same stimuli also elicited EEG and LFP responses, dominated by a biphasic wave that predicted the magnitude of the behavioral adjustment. These results disclose a phylogenetically preserved corticomotor mechanism supporting adaptive behavior in response to unexpected events.

Novel operator-splitting methods for real-time dynamic substructure testing

Improved and corrected Operator-Splitting (OS) methods are proposed for Real-Time Dynamic Substructure Testing (RTDST) in this paper. Firstly, the theory and the stability of two proposed OS methods are investigated. Furthermore, the effectiveness of two proposed OS methods is validated by RTDSTs with one pure mass specimen and one mass-stiffness specimen. By using the predictor velocity and acceleration as well as the correction force approximations, the improved OS method and the corrected OS method are explicit methods for RTDST. It is shown from the analytical and experimental results that the improved OS method is unconditionally stable for linear type and softening type stiffness and damping systems and conditionally stable for hardening type stiffness and damping system if the mass ratio of the experimental substructure to the numerical substructure is less than 1. In contrast, the improved OS method is unsuitable for RTDST if the mass ratio is larger than 1. The corrected OS method is unconditionally stable for the linear type and softening type stiffness and damping systems and conditionally stable for hardening type stiffness and damping systems. The stability of the corrected OS method is better than the improved OS method, which indicates the advantage of the corrected OS method over the improved OS method.

Canada at a Crossroads

Rooted in Canada’s history as a British colony, the Criminal Code defence to “reasonable corrective force” remains in force today. In this article, I trace the history of Canada’s law, describe various attempts made to remove it, and present the many recommendations and calls for repeal issued over recent decades. While Canada has stood still, 65 countries have removed their legal defences and prohibited all corporal punishment of children. At this time, Canada has another opportunity to finally remove this archaic defence, as two Private Members’ Bills have been introduced in the House of Commons and the Senate, respectively. By seizing this opportunity, Canada will demonstrate its commitments to children’s rights to protection and dignity, and uphold its pledges to end violence against children and women in Canada.

Biomechanical analysis of rod contouring in posterior spinal instrumentation and fusion for 3D correction of adolescent idiopathic scoliosis.

To biomechanically evaluate 3D corrective forces and deformity correction attributable to key parameters of rod contouring in posterior spinal fusion (PSF) for adolescent idiopathic scoliosis (AIS). Computerised patient-specific biomechanical models of six AIS cases were used to simulate PSF and evaluate the effects of 5.5-mm cobalt-chrome rod contouring angle (concave-convex angles: 30°-15°, 45°-15° and 60°-15°), length (spanning 4 and 7 vertebrae), and apex location (T7, T9). 3D correction and bone-implant forces were computed and analysed. By increasing the concave rod contour from 30° to 60°, thoracic kyphosis (TK) increased from 18° ± 2° (15°-19°) to 24° ± 2° (22°-26°), apical vertebra rotation (AVR) correction increased from 41%(SD8%) to 66%(SD18%) whilst the main thoracic curve (MT) correction decreased from 68%(SD6%) to 56%(SD8%). With a contouring length of 4 vs. 7 vertebrae, the resulting TK, AVR and MT corrections were 22° ± 1° (19°-26°) vs. 19° ± 10° (15°-22°), 57%(SD18%) vs. 50%(SD26%) and 59%(SD1%) vs. 69%(SD35%), respectively. With the rod contouring apex at T7 (vs. T9), AVR corrections were 69% (SD19%) vs. 44% (SD9%), with no significant difference in TK and MT corrections, and with comparatively 67% of screw pull-out forces. Corrective forces were more evenly shared with fixation on 7 vs. 4 vertebrae. Rod contouring of a greater angulation, over a shorter portion of the rod, and more centred at the apex of the main thoracic curve apex improved AVR correction and allowed greater restoration of TK, but resulted in significantly higher screw pull-out forces and came at the expense of less coronal plane correction.

Hydrodynamic aggregation of membrane inclusions due to non-Newtonian surface rheology

Biological membranes are self-assembled complex fluid interfaces that host proteins, molecular motors, and other macromolecules essential for cellular function. These membranes have a distinct in-plane fluid response with a surface viscosity that has been well characterized. The resulting quasi-two-dimensional fluid dynamical problem describes the motion of embedded proteins or particles. However, the viscous response of biological membranes is often non-Newtonian: in particular, the surface shear viscosity of phospholipids that comprise the membrane depends strongly on the surface pressure. We use the Lorentz reciprocal theorem to extract the effective long-ranged hydrodynamic interaction among membrane inclusions that arises due to such non-trivial rheology. We show that the corrective force that emerges ties back to the interplay between membrane flow and non-constant viscosity, which suggests a mechanism for biologically favorable protein aggregation within membranes. We quantify and describe the mechanism for such a large-scale concentration instability using a mean-field model. Finally, we employ numerical simulations to demonstrate the formation of hexatic crystals due to the effective hydrodynamic interactions within the membrane.

Optimization of in-brace corrective force in adolescents with Lenke type 5 curve using finite element model

BackgroundPelvic parameters have been taken into consideration for the evaluation of the outcomes of bracing in AIS. To discuss the stress required to correct the pelvic deformity related to Lenke5 adolescent idiopathic scoliosis (AIS) by finite element analysis, and provide a reference for the shaping of the pelvic region of the brace.MethodsAn three-dimensional (3D) corrective force on the pelvic area was defined. Computed tomography images were used to reconstruct a 3D model of Lenke5 AIS. Computer-aided engineering software Abaqus was used to implement finite element analysis. By adjusting the magnitude and position of corrective forces, coronal pelvic coronal plane rotation (PCPR) and Cobb angle (CA) of lumbar curve in the coronal plane, horizontal pelvic axial plane rotation, and apical vertebra rotation (AVR) were minimized to achieve the best effect on the spine and pelvic deformity correction. The proposed corrective conditions were divided into three groups: (1) forces applied on X-axis; (2) forces applied both in the X- and Y-axis; and (3) forces applied along the X-, Y-, and Z-axis at the same time.ResultsIn three groups, CA correction reduced by 31.5%, 42.5%, and 59.8%, and the PCPR changed to 12°, 13°, and 1° from 6.5°, respectively. The best groups of correction forces should simultaneously locate on the sagittal, transverse, and coronal planes of the pelvis.ConclusionsFor Lenke5 AIS, 3D correction forces can sufficiently reduce scoliosis and pelvic asymmetrical state. Force applied along the Z-axis is vital to correct the pelvic coronal pelvic tilt associated with Lenke5 AIS.

Complex-variable, high-precision formulation of the consistent boundary element method for 2D potential and elasticity problems

The collocation boundary element method was recently entirely revisited on the basis of a consistent derivation of Somigliana’s identity in terms of weighted residuals. Both conceptually and for the sake of code implementation, the correct traction force interpolation along generally curved boundaries, as for elasticity problems, leads to the enunciation of an inedited, actually long-sought, convergence theorem as well as to considerable numerical simplifications. Numerical evaluations for two-dimensional problems require exclusively Gauss–Legendre quadrature plus eventual correction terms obtained analytically regardless of the order or shape of the implemented boundary element interpolation. Arbitrarily high precision and accuracy is achievable for low-cost computation, as eventual mesh-subdivision refinements should take place only if the mechanical simulation demands — and not just for numerical evaluations. We now show that considerable simplification is obtained by switching the formulation from real to complex variable. Precision, round-off errors, and accuracy of a given numerical implementation may be kept – identifiably and separately – under control, as assessed for some potential and elasticity examples with extremely challenging topologies. In fact, source-field distances may be arbitrarily small — far smaller than deemed feasible in continuum mechanics, as we resort to nothing else than the problem’s mathematics.