Local Arc Length Research Articles

Plunging breakers. Part 1. Analysis of an ensemble of wave profiles

An experimental study of the dynamics and droplet production in three mechanically generated plunging breaking waves is presented in this two-part paper. In the present paper (Part 1), the dynamics of the three breakers are studied through measurements of the evolution of their free surface profiles during 10 repeated breaking events for each wave. The waves are created from dispersively focused wave packets generated with three wave maker motions that differ primarily by small changes in their overall amplitude. Breaker profiles are measured with a cinematic laser-induced fluorescence technique covering a streamwise region of approximately one breaker wavelength and over a time of 3.4 breaker periods. The aligned profile data is used to create spatio-temporal maps of the ensemble average surface height and the standard deviation of both the local normal distance and the local arc length relative to the instantaneous mean profile. It is found that the mean and standard deviation maps contain strongly correlated localized features and indicate that the transition from laminar to turbulent flow is a highly repeatable process. Regions of high standard deviation include the splash created by the plunging jet impact and subsequent splash impacts at the front of the breaking region as well as the site where the air pocket entrained under the plunging jet at the moment of jet tip impact comes to the surface and pops. In Part 2 (Erinin et al., J. Fluid Mech., vol. 967, 2023, A36), these features are used to interpret various features of the distributions of droplet number, diameter and velocity.

Large deflections of inextensible cantilevers: modeling, theory, and simulation

A recent large deflection cantilever model is considered. The principal nonlinear effects come through the beam’sinextensibility– local arc length preservation – rather than traditional extensible effects attributed to fully restricted boundary conditions. Enforcing inextensibility leads to:nonlinear stiffnessterms, which appear as quasilinear and semilinear effects, as well asnonlinear inertiaeffects, appearing as nonlocal terms that make the beam implicit in the acceleration. In this paper we discuss the derivation of the equations of motionviaHamilton’s principle with a Lagrange multiplier to enforce theeffective inextensibility constraint. We then provide the functional framework for weak and strong solutions before presenting novel results on the existence and uniqueness of strong solutions. A distinguishing feature is that the two types of nonlinear terms present independent challenges: the quasilinear nature of the stiffness forces higher topologies for solutions, while the nonlocal inertia requires the consideration of Kelvin-Voigt type damping to close estimates. Finally, a modal approach is used to produce mathematically-oriented numerical simulations that provide insight into the features and limitations of the inextensible model.

Isometric bending requires local constraints on free edges

While the shape equations describing the equilibrium of an unstretchable thin sheet that is free to bend are known, the boundary conditions that supplement these equations on free edges have remained elusive. Intuitively, unstretchability is captured by a constraint on the metric within the bulk. Naïvely one would then guess that this constraint is enough to ensure that the deformations determining the boundary conditions on these edges respect the isometry constraint. If matters were this simple, unfortunately, it would imply unbalanced torques (as well as forces) along the edge unless manifestly unphysical constraints are met by the boundary geometry. In this article, we identify the source of the problem: not only the local arc-length but also the geodesic curvature need to be constrained explicitly on all free edges. We derive the boundary conditions which follow. In contrast to conventional wisdom, there is no need to introduce boundary layers. This framework is applied to isolated conical defects, both with deficit as well, but more briefly, as surplus angles. Using these boundary conditions, we show that the lateral tension within a circular cone of fixed radius is equal but opposite to the radial compression, and independent of the deficit angle itself. We proceed to examine the effect of an oblique outer edge on this cone perturbatively demonstrating that both the correction to the geometry as well as the stress distribution in the cone kicks in at second order in the eccentricity of the edge.

Extraction Method of Coronary Artery Blood Vessel Centerline in CT Coronary Angiography

Aiming at the problem that the traditional coronary artery centerline extraction method is computationally intensive and requires a large number of manual interventions, a fully automatic coronary artery centerline tracking extraction method is proposed. First, adaptive adaptation of coronary vessels was performed based on different Fresenius norms designed for coronary scale. A multi-scale differential operator is constructed based on the vascular gray-scale distribution to establish a discriminant function, thereby obtaining the initial position and tracking direction of the seed point. Then, the new ridge point and tracking direction are optimally detected within the local arc length of the initial direction to ensure that the correct ridge position can still be obtained with large error interference. Finally, the pseudo vascular centerline is removed using the vascular topology feature detection method. The experimental results show that the method can accurately extract the blood vessel center line, direction vector and other information in the coronary angiography image without manual intervention, and can be used in the computer-assisted diagnosis and treatment process of clinical cardiovascular disease.

Stability analysis of slope in strain-softening soils using local arc-length solution scheme

Soils with strain-softening behavior — manifesting as a reduction of strength with increasing plastic strain — are commonly found in the natural environment. For slopes in these soils, a progressive failure mechanism can occur due to a reduction of strength with increasing strain. Finite element method based numerical approaches have been widely performed for simulating such failure mechanism, owning to their ability for tracing the formation and development of the localized shear strain. However, the reliability of the currently used approaches are often affected by poor convergence or significant mesh-dependency, and their applicability is limited by the use of complicated soil models. This paper aims to overcome these limitations by developing a finite element approach using a local arc-length controlled iterative algorithm as the solution strategy. In the proposed finite element approach, the soils are simulated with an elastoplastic constitutive model in conjunction with the Mohr-Coulomb yield function. The strain-softening behavior is represented by a piece-wise linear relationship between the Mohr-Coulomb strength parameters and the deviatoric plastic strain. To assess the reliability of the proposed finite element approach, comparisons of the numerical solutions obtained by different finite element methods and meshes with various qualities are presented. Moreover, a landslide triggered by excavation in a real expressway construction project is analyzed by the presented finite element approach to demonstrate its applicability for practical engineering problems.

Curve length estimation based on cubic spline interpolation in gray-scale images

This paper deals with a novel local arc length estimator for curves in gray-scale images. The method first estimates a cubic spline curve fit for the boundary points using the gray-level information of the nearby pixels, and then computes the sum of the spline segments’ lengths. In this model, the second derivatives and y coordinates at the knots are required in the computation; the spline polynomial coefficients need not be computed explicitly. We provide the algorithm pseudo code for estimation and preprocessing, both taking linear time. Implementation shows that the proposed model gains a smaller relative error than other state-of-the-art methods.

On meromorphic parameterizations of real algebraic curves

A singular flat geometry may be canonically assigned to a real algebraic curve Γ; namely, via analytic continuation of unit speed parameterization of the real locus \({\Gamma_\mathbb{R}}\) . Globally, the metric \({\rho=|Q|=|q(z)|dzd\bar{z}}\) is given by the meromorphic quadratic differential Q on Γ induced by the standard complex form dx2 + dy2 on \({\mathbb{C}^2=\{(x,y)\}}\) . By considering basic properties of Q, we show that the condition for local arc length parameterization along \({\Gamma_\mathbb{R}}\) to extend meromorphically to the complex plane is quite restrictive: For curves of degree at most four, only lines, circles and Bernoulli lemniscates have such meromorphic parameterizations.

Fully-automatic modelling of cohesive crack growth using a finite element–scaled boundary finite element coupled method

This study develops a method coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for fully-automatic modelling of cohesive crack growth in quasi-brittle materials. The simple linear elastic fracture mechanics (LEFM)-based remeshing procedure developed previously is augmented by inserting nonlinear interface finite elements automatically. The constitutive law of these elements is modelled by the cohesive/fictitious crack model to simulate the fracture process zone, while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. The crack is assumed to grow when the mode-I stress intensity factor K I vanishes in the direction determined by LEFM criteria. Other salient algorithms associated with the SBFEM, such as mapping state variables after remeshing and calculating K I using a “shadow subdomain”, are also described. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the new method. The results show that this SBFEM–FEM coupled method is capable of fully-automatically predicting both satisfactory crack trajectories and accurate load–displacement relations with a small number of degrees of freedom, even for problems with strong snap-back. Parametric studies were carried out on the crack incremental length, the concrete tensile strength, and the mode-I and mode-II fracture energies. It is found that the K I ⩾ 0 criterion is objective with respect to the crack incremental length.

Modelling cohesive crack growth using a two-step finite element-scaled boundary finite element coupled method

A two-step method, coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM), is developed in this paper for modelling cohesive crack growth in quasi-brittle normal-sized structures such as concrete beams. In the first step, the crack trajectory is fully automatically predicted by a recently-developed simple remeshing procedure using the SBFEM based on the linear elastic fracture mechanics theory. In the second step, interfacial finite elements with tension-softening constitutive laws are inserted into the crack path to model gradual energy dissipation in the fracture process zone, while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the proposed method. The numerical results demonstrate that this two-step SBFEM-FEM coupled method can predict both satisfactory crack trajectories and accurate load-displacement relations with a small number of degrees of freedom, even for crack growth problems with strong snap-back phenomenon. The effects of the tensile strength, the mode-I and mode-II fracture energies on the predicted load-displacement relations are also discussed.

A comparative study of numerical solutions to non-linear discrete crack modelling of concrete beams involving sharp snap-back

Numerical problems are often encountered in modelling crack propagation in concrete beams using non-linear finite element (FE) analysis, especially when sharp snap-back behaviour in load–displacement relations occurs. This paper firstly identifies 16 arc-length control based numerical strategies based on extensive literature review. They are then used to carefully model the structural behaviour of a four-point single notched shear beam using discrete crack modelling approach in which cracks are represented by interface elements with bilinear softening constitutive laws. Based on extensive FE analyses, detailed comparisons of the merits and demerits of these numerical algorithms are then made. The results indicate that the effectiveness and efficiency of different algorithms may vary considerably from one to another, with the local arc-length based procedures in conjunction with tangential stiffness strategy and reversible unloading model being the most robust.

A Front Dynamics Approach to Curvature-Dependent Flow

A front dynamics approach is developed to study the evolution of planar curves whose normal speed depends on curvature. The formulation is similar to Whitham’s shock dynamics theory for the propagation of shock wave in gases but assumes a different propagation rule. Equations that describe the motion of the front are obtained, and these are evolution equations for the normal direction and local arc length of the front. The solution of these equations leads to the front positions using an appropriate integration along rays. A similarity solution of the equations is found for the evolution of an initial corner. Free-boundary problems for the motion of a junction connecting front segments are discussed. A numerical method is presented to calculate the evolution of any number of front segments. The segments can be closed or open, connected to wall boundaries or not, or connected to other segments at 3-segment junctions. Several sample problems are considered to illustrate the method. An extension of the method for curvature-dependent motion under a constant area constraint is also discussed.