Simulating phase transformations by a meshless method with front tracking

A second order front tracking method is developed for solving the Euler equations of inviscid fluid dynamics numerically. Front tracking methods are usually limited to first order accuracy, since they are based on a piecewise constant approximation of the solution. Here the second order convergence is achieved by building a piecewise linear reconstruction of the piecewise constant front tracking solution in a post-processing step. The linearization is performed by decomposing the piecewise constant solution of the hyperbolic system into its wave components and by linearizing the wave solutions separately. In order to achieve a physically correct linearization, the front types of the previously developed improved front tracking method are employed. It is illustrated numerically for the one-dimensional unsteady interacting blast waves problem and a two-dimensional supersonic airfoil flow validation study that the proposed front tracking method can achieve second order convergence also in the presents of strong discontinuities.

A new front tracking scheme for the ultra-relativistic Euler equations

A two-dimensional front tracking method based on conservation for scalar conservation law and for the Euler system of gas dynamics is being developed. In the method, discontinuities are tracked by enforcing the conservation properties of the PDEs. Unlike the traditional front tracking methods, for which the conservation is only a property that may or may not be preserved, in this front tracking method the conservation is the mechanism by which the tracking is performed. The method is also extended to treat the reflection wall boundary in the Euler system. As an attempt to treat the interactions of discontinuities by enforcing the conservation properties without solving two-dimensional Riemann problems, three special cases of treatment of discontinuity interactions in scalar conservation law and in the Euler system of gas dynamics are studied. Finally, numerical experiments are implemented in both scalar and system cases to show the efficiency of the method.

The Hot Optimal Transportation Meshfree (HOTM) method for materials under extreme dynamic thermomechanical conditions

A note on front tracking for the Keyfitz–Kranzer system

An improved Front-Tracking technique for the simulation of mass transfer in dense bubbly flows

A new 3D third Order Front-Tracking method and its application to 2D flow with phase change is presented. The originality of this method lies in the meshing of the computational domain it generates: At each time step, the highly deformed interface is a grid surface. This allows an accurate computation of the physical field in the vicinity of the interface as an exact quantification of their eventual jumps at the interface. To step toward the understanding of the physical processes involved in liquid film evaporation, the 2D formulation of this new front tracking method is used by merging it in a numerical model solving the conservation equations in a quadratic finite element formulation. The heat transport equation is solved in the three different phases (solid, liquid and vapor) as the Navier-Stokes equations are solved in the two fluids. The temperature gradient discontinuity at the liquid vapor interface provides local value of the evaporative flux density that is directly linked to the interface velocity jump through mass conservation principle and used as boundary conditions for the two fluid flow computations. Testing on academic cases and application to axysimmetric film evaporation including comparison with experiments are shown.

A lattice Boltzmann front-tracking method for interface dynamics with surface tension in two dimensions

These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: (1) Meaning of a conservation equation and definition of weak solutions. (2) Hyperbolic systems. Explicit solutions in the linear, constant coefficients case. Nonlinear effects: loss of regularity and wave interactions. (3) Shock waves: Rankine–Hugoniot equations and admissibility conditions. (4) Genuinely nonlinear and linearly degenerate characteristic fields. Centered rarefaction waves. The general solution of the Riemann problem. Wave interaction estimates. (5) Weak solutions to the Cauchy problem, with initial data having small total variation. Approximations generated by the front-tracking method and by the Glimm scheme. (6) Continuous dependence of solutions w.r.t. the initial data, in the L 1 distance. (7) Characterization of solutions which are limits of front tracking approximations. Uniqueness of entropy-admissible weak solutions. (8) Vanishing viscosity approximations. (9) Extensions and open problems. The survey is concluded with an Appendix, reviewing some basic analytical tools used in the previous sections.Throughout the exposition, technical details are mostly left out. The main goal of these notes is to convey basic ideas, also with the aid of a large number of figures.KeywordsWeak SolutionCauchy ProblemHyperbolic SystemRiemann ProblemPiecewise Constant FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

The application of a front tracking method to a nonstrictly hyperbolic system of conservation laws is described in one space dimension. The front tracking method is based on approximate solutions of Riemann problems. The method is compared with the random choice scheme and the upwind scheme.

Anisotropic motion laws play a key role in many applications ranging from materials science, biophysics to image processing. All these highly diversified disciplines have made it necessary to develop common math- ematical foundations and framworks to deal with anisotropy in geometric motion. The workshop brings together leading experts from various fields to address well-posedness, accuracy, and computational efficiency of the math- ematical models and algorithms.

Three-dimensional numerical simulation of axis-switching and micro-droplet formation in a co-flowing immiscible elliptic jet flow system using front tracking method

A variety of particle-based methods have been developed for the purpose of computationally modelling processes that involve, for example, complex topological changes of interfaces, significant plastic deformation of materials, fluid flow in conjunction with heat transfer and phase transformation, flow in porous media, granular flow, etc. Being different from the conventional methods that directly solve related governing equations using a computational grid, the particle-based methods firstly discretize the continuous medium into discrete pseudo-particles in mathematics. The methods then mathematically solve the governing equations by considering the local interaction between neighbouring pseudo-particles. Such solutions can reflect the overall flow, deformation, heat transfer and phase transformation processes of the target materials at the mesoscale and macroscale. This paper reviews the fundamental concepts of four different particle-based methods (lattice Boltzmann method—LBM, smoothed particle hydrodynamics—SPH, discrete element method—DEM and particle finite element method—PFEM) and their application in computational modelling research on welding, casting and additive manufacturing.

Meshless methods (MMs) have become interesting and promising methods in solving partial differential equations, because of their flexibility in practical applications when compared with the standard finite-element method (e.g. crack propagation, large deformations, and so on). Implementation of these methods requires a good understanding of the influence of some specific selectable parameters. In this article, those parameters are analysed for one of the most popular MMs, the element-free Galerkin method, considering both accuracy and computational cost. Thus, the dependence of the solutions on grid irregularity, order of the polynomial basis, type of weight function, and the support size is investigated, and conclusions are drawn with respect to recommended or ‘optimal’ values for one-dimensional bar problems with applied axial loads.