1.1. Eulerian and Lagrangian coordinates. Let us begin with Eulerian and Lagrangian coordinates. The Eulerian coordinate (x, t) is the physical space plus time. The Eulerian description of the flow is to describe the flow using quantities as a function of a spatial location x and time t, e.g. the flow velocity u(x, t). This can be visualized by sitting on the bank of a river and watching the wa...

In this paper, we consider the adaptive Eulerian–Lagrangian method (ELM) for linear convection-diffusion problems. Unlike the classical a posteriori error estimations, we estimate the temporal error along the characteristics and derive a new a posteriori error bound for ELM semi-discretization. With the help of this proposed error bound, we are able to show the optimal convergence rate of ELM f...

Title of dissertation: CAPTURING MICRO-EMULSIONS AND MICRO-FOAMS WITH THE ARBITRARY LAGRANGIAN EULERIAN METHOD Andrew Brandon, Doctor of Philosophy, 2015 Dissertation directed by: Professor Konstantina Trivisa Department of Mathematics The goal of this work is to develop a 2D, heterogenous model for the purpose of studying liquid drainage from micro-foams and micro-emulsions. Liquid drainage wi...

The Eulerian–Lagrangian method of fundamental solutions is proposed to solve the two-dimensional unsteady Burgers’ equations. Through the Eulerian–Lagrangian technique, the quasi-linear Burgers’ equations can be converted to the characteristic diffusion equations. The method of fundamental solutions is then adopted to solve the diffusion equation through the diffusion fundamental solution; in t...

ALEGRA is a multi-material, arbitrary-Lagrangian-Eulerian (ALE) code for solid dynamics being developed by the Computational Physics Research and Development Department at Sandia National Laboratories. It combines the features of modern Eulerian shock codes, such as CTH, with modern Lagrangian structural analysis codes. With the ALE algorithm, the mesh can be stationary (Eulerian) with the mate...

A coordinate independent derivation of the Eulerian and Lagrangian strain tensors of finite deformation theory is given based on the parallel propagator, the world function, and the displacement vector field as a three-point tensor. The derivation explicitly shows that the Eulerian and Lagrangian strain tensors are two-point tensors, each a function of both the spatial and material coordinates....

In this work, the existence of a global minimizer for the previous Lagrangian formulation of nonpolar solvationmodel proposed in [1] has been proved. One of the proofs involves a construction of a phase field model that converges to the Lagrangian formulation. Moreover, an Eulerian formulation of nonpolar solvationmodel is proposed and implemented under a similar parameterization scheme to that...

We present a method for applying semi-implicit forces on a Lagrangian mesh to an Eulerian discretization of the Navier Stokes equations in a way that produces a sparse symmetric positive definite system. The resulting method has semi-implicit and fully-coupled viscosity, pressure, and Lagrangian forces. We apply our new framework for forces on a Lagrangian mesh to the case of a surface tension ...

We present a general Lagrangian formalism that allows the treatment of vorticity. We give solutions for the rotational perturbations up to the third-order in a flat background universe. We show how the primordial vorticity affects the evolution of the density fluctuation in high-density regions. The study of the galaxy formation based on gravitational instability has not revealed the origin of ...

A closed set of exact equations describing statistical theory of turbulent self-diffusion by multivariate-normal turbulent velocity field is derived. In doing so, we first suggest exact formulas for correlations 〈fi(p)fj(p ′)R[f ]〉, 〈g(p)R[f ]〉 and 〈g(p)fj(p ′)R[f ]〉 when the functional R[f ] is functional of functions fi’s having multivariate-normal distribution, g and fi’s have joint normal d...