In this article, we detail the methodology developed to construct arbitrarily high order schemes — linear and WENO — on 3D mixed-element unstructured meshes made up of general convex polyhedral elements. The approach is tailored specifically for the solution of scalar level set equations for application to incompressible two-phase flow problems. The construction of WENO schemes on 3D unstructured meshes is notoriously difficult, as it involves a much higher level of complexity than 2D approaches. This due to the multiplicity of geometrical considerations introduced by the extra dimension, especially on mixed-element meshes. Therefore, we have specifically developed a number of algorithms to handle mixed-element meshes composed of convex polyhedra with convex polygonal faces. The contribution of this work concerns several areas of interest: the formulation of an improved methodology in 3D, the minimisation of computational runtime in the implementation through the maximum use of pre-processing operations, the generation of novel methods to handle complex 3D mixed-element meshes and finally the application of the method to the transport of a scalar level set.

The most severe limitation of the standard Lattice Boltzmann Method is the use of uniform Cartesian grids especially when there is a need for high resolutions near the body or the walls. Among the recent advances in lattice Boltzmann research to handle complex geometries, a particularly remarkable option is represented by changing the solution procedure from the original "stream and collide" to a finite volume technique. However, most of the presented schemes have stability problems. This paper presents a stable and accurate finite-volume lattice Boltzmann formulation based on a cell-centred scheme. To enhance stability, upwind second order pressure biasing factors are used as flux correctors on a D2Q9 lattice. The resulting model has been tested against a uniform flow past a cylinder and typical free shear flow problems at low and moderate Reynolds numbers: boundary layer, mixing layer and plane jet flows. The numerical results show a very good accuracy and agreement with the exact solution of the Navier-Stokes equation and previous numerical results and/or experimental data. Results in self-similar coordinates are also investigated and show that the time-averaged statistics for velocity and vorticity express self-similarity at low Reynolds numbers. Furthermore, the scheme is applied to simulate the flow around circular cylinder and the Reynolds number range is chosen in such a way that the flow is time dependent. The agreement of the numerical results with previous results is satisfactory.

We couple different flow models, i.e. a finite element solver for the Navier-Stokes equations and a Lattice Boltzmann automaton, using the framework Peano as a common base. The new coupling strategy between the meso- and macroscopic solver is presented and validated in a 2D channel flow scenario. The results are in good agreement with theory and results obtained in similar works by Latt et al. In addition, the test scenarios show an improved stability of the coupled method compared to pure Lattice Boltzmann simulations.

The present paper provides a numerical investigation of the decoherence effect induced on a quantum heavy particle by the scattering with a light one. The time dependent two-particle Schrödinger equation is solved by means of a time-splitting method. The damping undergone by the non-diagonal terms of the heavy particle density matrix is estimated numerically, as well as the error in the Joos-Zeh approximation formula. 

We consider a non-standard mixed method for the Stokes problem in Rⁿ, n∈{2,3}, with Dirichlet boundary conditions, in which, after using the incompressibility condition to eliminate the pressure, the pseudostress tensor σ and the velocity vector u become the only unknowns. Then, we apply the Babuška-Brezzi theory to prove the well-posedness of the corresponding continuous and discrete formulations. In particular, we show that Raviart-Thomas elements of order k≥0 for σ and piecewise polynomials of degree k for u ensure unique solvability and stability of the associated Galerkin scheme. In addition, we introduce and analyze an augmented approach for our pseudostress-velocity formulation. The methodology employed is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and the Dirichlet boundary condition for the velocity, all of them multiplied by suitable stabilization parameters. We show that these parameters can be chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. For instance, Raviart-Thomas elements of order k≥0 for σ and continuous piecewise polynomials of degree k+1 for u become a feasible choice in this case. Finally, become a feasible choice in this case. Finally, extensive numerical experiments illustrating the good performance of the methods and comparing them with other procedures available in the literature, are provided.

Orbital-free density functional theory (OFDFT) is a quantum mechanical method in which the energy of a material depends only on the electron density and ionic positions. We examine some popular algorithms for optimizing the electron density distribution in OFDFT, explaining their suitability, benchmarking their performance, and suggesting some improvements. We start by describing the constrained optimization problem that encompasses electron density optimization. Next, we discuss the line search (including Wolfe conditions) and the nonlinear conjugate gradient and truncated Newton algorithms, as implemented in our open source OFDFT code. We finally focus on preconditioners derived from OFDFT energy functionals. Newly-derived preconditioners are successful for simulation cells of all sizes without regions of low electron-density and for small simulation cells with such regions.

The paper is devoted to the development of an efficient deterministic framework for modelling of three-dimensional rarefied gas flows on the basis of the numerical solution of the Boltzmann kinetic equation with the model collision integrals. The framework consists of a high-order accurate implicit advection scheme on arbitrary unstructured meshes, the conservative procedure for the calculation of the model collision integral and efficient implementation on parallel machines. The main application area of the suggested methods is micro-scale flows. Performance of the proposed approach is demonstrated on a rarefied gas flow through the finite-length circular pipe. The results show good accuracy of the proposed algorithm across all flow regimes and its high efficiency and excellent parallel scalability for up to 512 cores.

We develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902–1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

In this paper, we develop an efficient numerical method based on the boundary integral equation formulation and new version of fast multipole method to solve the boundary value problem for the stress field associated with dislocations in a finite medium. Numerical examples are presented to examine the influence from material boundaries on dislocations.

The concept of diffusion regulation (DR) was originally proposed by Jaisankar for traditional second order finite volume Euler solvers. This was used to decrease the inherent dissipation associated with using approximate Riemann solvers. In this paper, the above concept is extended to the high order spectral volume (SV) method. The DR formulation was used in conjunction with the Rusanov flux to handle the inviscid flux terms. Numerical experiments were conducted to compare and contrast the original and the DR formulations. These experiments demonstrated (i) retention of high order accuracy for the new formulation, (ii) higher fidelity of the DR formulation, when compared to the original scheme for all orders and (iii) straightforward extension to Navier Stokes equations, since the DR does not interfere with the discretization of the viscous fluxes. In general, the 2D numerical results are very promising and indicate that the approach has a great potential for 3D flow problems.

In this paper we present a selective segmentation model using a dual level set variational formulation. Our variational model aims to segment all objects with one level set function (global) and the selected object, which is the closest to the geometric constraints (markers), with another level set (local). It is a combination of edge detection, markers distance function and active contour without edges. Experimental results show that our model is more robust than previous work. 

A concept of "static reconstruction" and "dynamic reconstruction" was introduced for higher-order (third-order or more) numerical methods in our previous work. Based on this concept, a class of hybrid DG/FV methods had been developed for one-dimensional conservation law using a "hybrid reconstruction" approach, and extended to two-dimensional scalar equations on triangular and Cartesian/triangular hybrid grids. In the hybrid DG/FV schemes, the lower-order derivatives of the piecewise polynomial are computed locally in a cell by the traditional DG method (called as "dynamic reconstruction"), while the higher-order derivatives are reconstructed by the "static reconstruction" of the FV method, using the known lower-order derivatives in the cell itself and in its adjacent neighboring cells. In this paper, the hybrid DG/FV schemes are extended to two-dimensional Euler equations on triangular and Cartesian/triangular hybrid grids. Some typical test cases are presented to demonstrate the performance of the hybrid DG/FV methods, including the standard vortex evolution problem with exact solution, isentropic vortex/weak shock wave interaction, subsonic flows past a circular cylinder and a three-element airfoil (30P30N), transonic flow past a NACA0012 airfoil. The accuracy study shows that the hybrid DG/FV method achieves the desired third-order accuracy, and the applications demonstrate that they can capture the flow structure accurately, and can reduce the CPU time and memory requirement greatly than the traditional DG method with the same order of accuracy.

An unstructured nodal spectral-element method for the Navier-Stokes equations is developed in this paper. The method is based on a triangular and tetrahedral rational approximation and an easy-to-implement nodal basis which fully enjoys the tensorial product property. It allows arbitrary triangular and tetrahedral mesh, affording greater flexibility in handling complex domains while maintaining all essential features of the usual spectral-element method. The details of the implementation and some numerical examples are provided to validate the efficiency and flexibility of the proposed method.