Discontinuous Velocity Fields Research Articles

A model of porous plastic single crystals based on fractal slip lines distribution

The ductile failure of crystalline materials is strongly linked to the growth of intragranular voids. The estimation of the overall yield criterion thus requires to take into account the anisotropic plastic behavior of the single crystal. In the framework of the kinematic limit-analysis approach, this problem has been considered up to now with Gurson-type isotropic trial velocity fields. In the present work, a different class of piecewise constant velocity fields is proposed based on a detailed analysis of FFT numerical results on the strain localization in porous single crystals with periodic distributions of voids. This original approach is implemented for the model 2D problem of a square or hexagonal array of cylindrical voids in a hexagonal close-packed single crystal with in-plane prismatic slip systems. For equibiaxial loadings, the assumption of discontinuous velocity field provides a good approximation of the smooth jumps observed in the numerical results. Consistently, this new proposal leads to a significant improvement on the macroscopic yield stress with respect to the estimate based on an isotropic velocity field. Our theoretical estimate almost coincides with the FFT numerical results for all the unit-cells and crystalline orientations considered.

A stabilizer-free pressure-robust finite element method for the Stokes equations

In this paper, we introduce a new finite element method for solving the Stokes equations in the primary velocity-pressure formulation using H(div) finite elements to approximate velocity. Like other finite element methods with velocity discretized by H(div) conforming elements, our method has the advantages of an exact divergence-free velocity field and pressure-robustness. However, most of H(div) conforming finite element methods for the Stokes equations require stabilizers to enforce the weak continuity of velocity in tangential direction. Some stabilizers need to tune penalty parameter and some of them do not. Our method is stabilizer free although discontinuous velocity fields are used. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. Extensive numerical investigations are conducted to test accuracy and robustness of the method and to confirm the theory. The numerical examples cover low- and high-order approximations up to the degree four, and 2D and 3D cases.

Hybrid Large Eddy Simulation for low-order Discontinuous Galerkin methods using an explicit filter

In this paper we present a simple, easily implemented and effective approach for explicitly-filtered Large Eddy Simulation with a Discontinuous Galerkin (DG) discretisation for velocity. DG formulations are often desirable due to their stability and increased accuracy, however this can come at greater computational expense due to the additional degrees of freedom in the velocity field. Additionally, data output can also be an issue, due to the increased storage requirements. Here we present a hybrid approach, based upon the construction of an approximation of the velocity shear tensor using information from a projected Continuous Galerkin (CG) version of the discontinuous velocity field. The resulting turbulence algorithm is implemented within Fluidity, an open-source computational fluid dynamics solver. The model is then validated with a well known test case, and shown to agree favourably with published results. Comparisons are also made between the CG/DG hybrid LES with DG-only LES, which demonstrate the superior computational performance of the hybrid model.

Representation of discontinuous seismic velocity fields by sigmoidal functions for ray tracing and traveltime modelling

SUMMARYWave-modelling methods based on asymptotic ray theory have a lower computational cost than full wave-equation methods but require a smooth velocity field, though discontinuities may be handled by imposing interface conditions between adjacent blocks. We propose to approximate discontinuous velocity fields with model parametrizations based on smooth, rapidly varying functions known as sigmoidal functions. We have implemented the proposed technique on Cartesian grids using the wavelet theory formalism. Numerical experiments with 2-D and 3-D initial-value and two-point ray tracing in heterogeneous media show that the ray paths and traveltimes produced with the sigmoidal representation are consistent with the results produced by conventional ray tracing in block structures, broadening the scope of classical algorithms based on smooth velocity fields.

Indirect extraction of bar stocks in viscous flow mode

The process of isothermal indirect extrusion is considered under conditions of viscoplastic metal flow, which provides the forming of high-strength nonferrous alloys. We believe that stamping is performed under conditions of strain hardening and fracture of the processed material. The deformation kinematics was established based on discontinuous path velocity fields. An expression is obtained for estimating the specific pressure during backward extrusion based on the upper boundary method. Dependencies were obtained showing the influence of the relative value of the tool stroke, reduction during extrusion, friction between the tool and the workpiece, the strain rate on the specific pressure during extrusion. The results can be used to assess the influence of technological parameters on the course of the operation.

Isothermal extrusion of ribs on plate

Technological schemes and relationships are given for calculating of operating modes during isothermal heating of workpieces. The conditions of viscoplastic non-stationary deformation are accepted. The energy form equilibrium equation as applied to the discontinuous displacement velocity fi eld and material damage kinetics equations are used. The calculation results and product samples are presented.

Sharp uniqueness conditions for one-dimensional, autonomous ordinary differential equations

We give two conditions that are necessary and sufficient for the uniqueness of Filippov solutions to scalar, autonomous ordinary differential equations with discontinuous velocity fields. When only one of the two conditions is satisfied, we give a natural selection criterion that guarantees uniqueness of the solution.

Transient modelling of flexible belt drive dynamics using the equations of a deformable string with discontinuities

ABSTRACTDynamics of a belt drive is analysed using a non-linear model of an extensible string at contour motion, in which the trajectories of particles of the belt are predetermined. The equations of string dynamics at the tight and slack spans are considered in a fixed domain by transforming into a spatial frame. Assuming the absence of slip of the belt on the surface of the pulleys, we arrive at a new model with a discontinuous velocity field and concentrated contact forces. Finite difference discretization allows numerical analysis of the resulting system of partial differential equations with delays. Example solution for the acceleration of a belt drive and an investigation of its frequency response depending on the velocity are presented and discussed.

Cohesion and suction induced hang-up in ore passes

This paper addresses the problem of hang-up in ore passes. A hang-up in a plane ore pass is analysed using the methods of discontinuous stress and velocity fields. The system consisting of broken ore material and a stiff ore pass wall is idealised as a plastic-rigid continuum satisfying the Mohr-Coulomb failure criterion and an associated flow rule. The influence of moisture in the ore, and its associated suction, is accounted for using the effective stress concept for unsaturated geomaterials. A simple failure mechanism consisting of three rigid blocks is assumed which is consistent with a bursting failure type and with the existence of a compressive arch underneath the hang-up section. Stress equilibrium and kinematic restrictions enable the derivation of simple algebraic expressions governing the stability of a hang-up. The separate influences of suction and cohesion are evident in the derived expressions. The paper then applies the derived expressions to a hang-up problem. Strength properties of an ore are determined experimentally using triaxial tests. The way water is retained in the ore is described using its fractal characteristics. It is shown how the ore pass size for hang-up instability is strongly dependant on moisture content and suction. An increase in suction increases the minimum ore pass dimension needed to avoid hang-up formation.

On Dynamics of Lagrangian Trajectories for Hamilton–Jacobi Equations

Characteristic curves of a Hamilton-Jacobi equation can be seen as action minimizing trajectories of fluid particles. However this description is valid only for smooth solutions. For nonsmooth "viscosity" solutions, which give rise to discontinuous velocity fields, this picture holds only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we discuss two physically meaningful regularisation procedures, one corresponding to vanishing viscosity and another to weak noise limit. We show that for any convex Hamiltonian, a viscous regularization allows to construct a nonsmooth flow that extends particle trajectories and determines dynamics inside the shock manifolds. This flow consists of integral curves of a particular "effective" velocity field, which is uniquely defined everywhere in the flow domain and is discontinuous on shock manifolds. The effective velocity field arising in the weak noise limit is generally non-unique and different from the viscous one, but in both cases there is a fundamental self-consistency condition constraining the dynamics.

Two-parameter Rigid Block Approach to Upper Bound Analysis of Equal Channel Angular Extrusion Through a Segal 2θ-die

This article deals with a phenomenological description of experimentally determined complex geometric shape of material dead zone during Equal Channel Angular Extrusion (ECAE) through a Segal 2θ-die with a channel intersection angle of 2θ>0°and 2θ<180°. Taking into account the complex dead zone geometry in a 2θ-die, a two-parameter Rigid Block Method (RBM) approach to a two-parameter Upper Bound Method (UBM) has been introduced with Discontinuous Velocity Field (DVF) for planar flow of plastic incompressible continua. The two-parameter UBM has allowed us to derive the numerical estimations for such energy-power parameters of ECAE as punching pressure and accumulated plastic strain for 2θ-dies. The obtained computational data have been compared with the one-parameter analytic UBM solution. Good agreement between the two computational results has been found.

Locking‐free discontinuous finite elements for the upper bound yield design of thick plates

SummaryThis work investigates the formulation of finite elements dedicated to the upper bound kinematic approach of yield design or limit analysis of Reissner–Mindlin thick plates in shear‐bending interaction. The main novelty of this paper is to take full advantage of the fundamental difference between limit analysis and elasticity problems as regards the class of admissible virtual velocity fields. In particular, it has been demonstrated for 2D plane stress, plane strain or 3D limit analysis problems that the use of discontinuous velocity fields presents a lot of advantages when seeking for accurate upper bound estimates. For this reason, discontinuous interpolations of the transverse velocity and the rotation fields for Reissner–Mindlin plates are proposed. The subsequent discrete minimization problem is formulated as a second‐order cone programming problem and is solved using the industrial software package MOSEK. A comprehensive study of the shear‐locking phenomenon is also conducted, and it is shown that discontinuous elements avoid such a phenomenon quite naturally whereas continuous elements cannot without any specific treatment. This particular aspect is confirmed through numerical examples on classical benchmark problems and the so‐obtained upper bound estimates are confronted to recently developed lower bound equilibrium finite elements for thick plates. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Presentation of the dynamical core of neXtSIM, a new sea ice model

The dynamical core of a new sea ice model is presented. It is based on the Elasto–Brittle rheology, which is inspired by progressive damage models used for example in rock mechanics. The main idea is that each element can be damaged when the local internal stress exceeds a Mohr–Coulomb failure criterion. The model is implemented with a finite element method and a Lagrangian advection scheme. Simulations of 10 days are performed over the Arctic at a resolution of 7 km. The model, which has only a few parameters, generates discontinuous sea ice velocity fields and strongly localized deformation features that occupy a few percent of the total sea ice cover area but accommodate most of the deformation. For the first time, a sea ice model is shown to reproduce the multifractal scaling properties of sea ice deformation. The sensitivity to model parameters and initial conditions is presented, as well as the ability of the Lagrangian advection scheme at preserving discontinuous fields.

Modeling shallow avalanche onset over complex basal topography

This paper deals with modeling of the onset of a shallow avalanche (soils, snow or other geomaterials) over various bottom topologies (mountains, valleys, ...). In order to do that, we use the shallow visco-plastic model with topography, developed in Ionescu (J. Non-Newtonian Fluid Mech. 193:116---128 2013), and we introduce a simple criterion to distinguish if an avalanche occurs or not. This criterion, relating the yield limit (material resistance) to the distribution of the external forces, is deduced from an optimization problem, called limit load analysis. The plastic dissipation functional which is involved in the limit load problem is non smooth and non coercive in the classical Sobolev spaces. To prove the existence of an onset velocity field (collapse flow) the appropriate functional space consists of bounded tangential deformation functions. We propose therefore a numerical strategy to solve the limit load problem and to get the onset flow field. A mesh free method, called the discontinuous velocity domain splitting (DVDS), is adapted here. The limit load problem is thus reduced to the minimization of a shape dependent functional. The discontinuous collapse flow velocity field is associated to a sub-domain and a rigid flow. With a level set of a Fourier function we give a description of the sub-domains and then we use genetic algorithms to solve the resulted non convex and non smooth global optimization problem. Finally, we illustrate the proposed numerical approach by solving several safety factor problems.

Modeling the Dynamics of a Flexible Belt Drive Using the Equations of a Deformable String with Discontinuities

The transient analysis of a belt drive is based on a nonlinear dynamic model of an extensible string at contour motion, in which the trajectories of particles of the belt are predetermined. The equations of string dynamics at the free spans are considered in a fixed domain by transforming into a spatial frame. Assuming the absence of slip of the belt on the surface of the pulleys, we arrive at a new model with a discontinuous velocity field and concentrated contact forces. Finite difference discretization allows numerical analysis of the resulting system of partial differential equations with delays. Example solution for the case of start up and accelerated motion of a friction belt drive is presented and discussed.

2D upper bound analysis of ECAE through 2θ-dies for a range of channel angles

Equal Channel Angular Extrusion (ECAE), sometimes referred to as Equal Channel Angular Pressing (ECAP), is a Severe Plastic Deformation technique. However the energy-power parameters of ECAE have not been fully addressed in previous known publications. The present article is focused on the punching pressure, accumulated plastic strain, and dead metal zone geometry estimation during ECAE of metal workpieces through a 2θ-die with a channel intersection angle of 2θ>0o and 2θ≠90o. Computational analytical results for ECAE technological parameters have been analytically derived for planar flow of a plastic, incompressible, non-hardening metal workpiece in an angular Segal die with 2θ>0o and 2θ≠90o. This is accomplished through the use of an Upper Bound Method (UBM) with Discontinuous Velocity Field (DVF) introduction. The development of the Dead Zone (DZ) for metal ECAE through a 2θ-die with 2θ>0o and 2θ≠90o has been analytically investigated. The obtained computational results for 2θ-die have been compared with the slip line analytic solutions of Segal for non-rectangular 2θ-dies of the same geometry. Good agreement between the two computational results has been found. The physical modeling techniques using plasticine have confirmed the appearance of a dead zone and material flow dynamics during ECAE through the Segal 2θ-die.

Limit analysis and lower/upper bounds to the macroscopic criterion of Drucker–Prager materials with spheroidal voids

The paper is devoted to a numerical Limit Analysis of a hollow spheroidal model with a Drucker–Prager solid matrix, for several values of the corresponding friction angle ϕ. In the first part of this study, the static and the mixed kinematic 3D-codes recently evaluated in [1] are modified to use the geometry defined in [2] for spheroidal cavities in the context of a von Mises matrix. The results in terms of macroscopic criteria are satisfactory for low and medium values of ϕ, but not enough for ϕ=30° in the highly compressive part of the criterion. To improve these results, an original mixed approach, dedicated to the axisymmetric case, was elaborated with a specific discontinuous quadratic velocity field associated with the triangular finite element. Despite the less good conditioning inherent to the axisymmetric modelization, the resulting conic programming problem appears quite efficient, allowing one take into account numerical discretization refinements unreachable with the corresponding 3D mixed code. After a first validation in the case of spherical cavities whose exact solution is known, the final results for spheroidal voids are given for three usual values of the friction angle and two values of the cavity aspect factor.

3D‐FEM formulations of limit analysis methods for porous pressure‐sensitive materials

SUMMARYThe first purpose of this paper is the numerical formulation of the three general limit analysis methods for problems involving pressure‐sensitive materials, that is, the static, classic, and mixed kinematic methods applied to problems with Drucker–Prager, Mises–Schleicher, or Green materials. In each case, quadratic or rotated quadratic cone programming is considered to solve the final optimization problems, leading to original and efficient numerical formulations. As a second purpose, the resulting codes are applied to non‐classic 3D problems, that is, the Gurson‐like hollow sphere problem with these materials as matrices. To this end are first presented the 3D finite element implementations of the static and kinematic classic methods of limit analysis together with a mixed method formulated to give also a purely kinematic result. Discontinuous stress and velocity fields are included in the analysis. The static and the two kinematic approaches are compared afterwards in the hydrostatic loading case whose exact solution is known for the three cases of matrix. Then, the static and the mixed approaches are used to assess the available approximate criteria for porous Drucker–Prager, Mises–Schleicher, and Green materials. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Application of Sobolev gradient techniques to two-point ray tracing

The Sobolev gradient steepest descent provides a numerically efficient iterative bending algorithm for two-point ray tracing. This new algorithm has advantages in its relative ease of numerical implementation, its ability to provide accurate solutions in discontinuous velocity fields (including head wave solutions), and its ability, when smooth velocity fields are used, to provide accurate solutions while using relatively few ray segments. The algorithm owes its advantages to its implicit nature, which it inherits from the Sobolev gradient solution method. For most models, the algorithm can be implemented such that at convergence, the norm of the gradient approaches zero. Thus, convergence implies a stationary point of the traveltime integral, guaranteeing that the rays found satisfy Fermat’s principle.

Explicit Dynamic Finite Element Method for Predicting Implosion/Explosion Induced Failure of Shell Structures

A simplified implementation of the conventional extended finite element method (XFEM) for dynamic fracture in thin shells is presented. Though this implementation uses the same linear combination of the conventional XFEM, it allows for considerable simplifications of the discontinuous displacement and velocity fields in shell finite elements. The proposed method is implemented for the discrete Kirchhoff triangular (DKT) shell element, which is one of the most popular shell elements in engineering analysis. Numerical examples for dynamic failure of shells under impulsive loads including implosion and explosion are presented to demonstrate the effectiveness and robustness of the method.