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.

Quantum molecular dynamic simulations have been employed to study the equation of state (EOS) of fluid helium under shock compressions. The principal Hugoniot is determined from EOS, where corrections from atomic ionization are added onto the calculated data. Our simulation results indicate that principal Hugoniot shows good agreement with gas gun and laser driven experiments, and maximum compression ratio of 5.16 is reached at 106 GPa.

The governing equations for heat transfer and fluid flow are often formulated in a general form for the simplification of discretization and programming, which has achieved great success in thermal science and engineering. Based on the analysis of the popular general form of governing equations, we found that energy conservation cannot be guaranteed when specific heat capacity is not constant, which may lead to unreliable results. A new concept of generalized density is put forward, based on which a new general form of governing equations is proposed to guarantee energy conservation. A number of calculation examples have been employed to verify validation and feasibility.

An iterative solver based on the immersed interface method is proposed to solve the pressure in a two-fluid flow on a Cartesian grid with second-order accuracy in the infinity norm. The iteration is constructed by introducing an unsteady term in the pressure Poisson equation. In each iteration step, a Helmholtz equation is solved on the Cartesian grid using FFT. The combination of the iteration and the immersed interface method enables the solver to handle various jump conditions across two-fluid interfaces. This solver can also be used to solve Poisson equations on irregular domains.

In this paper the numerical method for solution of an aeroelastic model describing the interactions of air flow with vocal folds is described. The flow is modelled by the incompressible Navier-Stokes equations spatially discretized with the aid of the stabilized finite element method. The motion of the computational domain is treated with the aid of the Arbitrary Lagrangian Eulerian method. The structure dynamics is replaced by a mechanically equivalent system with the two degrees of freedom governed by a system of ordinary differential equations and discretized in time with the aid of an implicit multistep method and strongly coupled with the flow model. The influence of inlet/outlet boundary conditions is studied and the numerical analysis is performed and compared to the related results from literature.

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.

A novel full Eulerian fluid-elastic membrane coupling method on the fixed Cartesian coordinate mesh is proposed within the framework of the volume-of-fluid approach. The present method is based on a full Eulerian fluid-(bulk) structure coupling solver (Sugiyama et al., J. Comput. Phys., 230 (2011) 596-627), with the bulk structure replaced by elastic membranes. In this study, a closed membrane is consid-ered, and it is described by a volume-of-fluid or volume-fraction information generally called VOF function. A smoothed indicator (or characteristic) function is introduced as a phase indicator which results in a smoothed VOF function. This smoothed VOF function uses a smoothed delta function, and it enables a membrane singular force to be incorporated into a mixture momentum equation. In order to deal with a membrane deformation on the Eulerian mesh, a deformation tensor is introduced and updated within a compactly supported region near the interface. Both the neo-Hookean and the Skalak models are employed in the numerical simulations. A smoothed (and less dissipative) interface capturing method is employed for the advection of the VOF function and the quantities defined on the membrane. The stability restriction due to membrane stiffness is relaxed by using a quasi-implicit approach. The present method is validated by using the spherical membrane deformation problems, and is applied to a pressure-driven flow with the biconcave membrane capsules (red blood cells).

There is a growing trend within energy and environmental simulation to consider tightly coupled solutions to multiphysics problems. This can be seen in nuclear reactor analysis where analysts are interested in coupled flow, heat transfer and neutronics, and in nuclear fuel performance simulation where analysts are interested in thermomechanics with contact coupled to species transport and chemistry. In energy and environmental applications, energy extraction involves geomechanics, flow through porous media and fractured formations, adding heat transport for enhanced oil recovery and geothermal applications, and adding reactive transport in the case of applications modeling the underground flow of contaminants. These more ambitious simulations usually motivate some level of parallel computing. Many of the physics coupling efforts to date utilize simple code coupling or first-order operator splitting, often referred to as loose coupling. While these approaches can produce answers, they usually leave questions of accuracy and stability unanswered. Additionally the different physics often reside on distinct meshes and data are coupled via simple interpolation, again leaving open questions of stability and accuracy.

A reconstruction-based discontinuous Galerkin (RDG(P1P2)) method, a variant of P1P2 method, is presented for the solution of the compressible Euler equations on arbitrary grids. In this method, an in-cell reconstruction, designed to enhance the accuracy of the discontinuous Galerkin method, is used to obtain a quadratic polynomial solution (P2) from the underlying linear polynomial (P1) discontinuous Galerkin solution using a least-squares method. The stencils used in the reconstruction involve only the von Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG(P1P2) method is third-order accurate, and outperforms the third-order DG method (DG(P2)) in terms of both computing costs and storage requirements.

Two-phase flow and heat transfer, such as boiling and condensing flows, are complicated physical phenomena that generally prohibit an exact solution and even pose severe challenges for numerical approaches. If numerical solution time is also an issue the challenge increases even further. We present here a numerical implementation and novel study of a fully distributed dynamic one-dimensional model of two-phase flow in a tube, including pressure drop, heat transfer, and variations in tube cross-section. The model is based on a homogeneous formulation of the governing equations, discretized by a high resolution finite difference scheme due to Kurganov and Tadmore.

The homogeneous formulation requires a set of thermodynamic relations to cover the entire range from liquid to gas state. This leads a number of numerical challenges since these relations introduce discontinuities in the derivative of the variables and are usually very slow to evaluate. To overcome these challenges, we use an interpolation scheme with local refinement.

The simulations show that the method handles crossing of the saturation lines for both liquid to two-phase and two-phase to gas regions. Furthermore, a novel result obtained in this work, the method is stable towards dynamic transitions of the inlet/outlet boundaries across the saturation lines. Results for these cases are presented along with a numerical demonstration of conservation of mass under dynamically varying boundary conditions. Finally we present results for the stability of the code in a case of a tube with a narrow section.

In this paper, we investigate the dynamic process of liquid bridge formation between two parallel hydrophobic plates with hydrophilic patches, previously studied in [1]. We propose a dynamic Hele-Shaw model to take advantage of the small aspect ratio between the gap width and the plate size. A constrained level set method is applied to solve the model equations numerically, where a global constraint is imposed in the evolution [2] stage together with local constraints in the reinitialization [3] stage of level set function in order to limit numerical mass loss. In contrast to the finite element method used in [2], we use a finite difference method with a 5th order HJWENO scheme for spatial discretization. To illustrate the effectiveness of the constrained method, we have compared the results obtained by the standard level set method with those from the constrained version. Our results show that the constrained level set method produces physically reasonable results while that of the standard method is less reliable. Our numerical results also show that the dynamic nature of the flow plays an important role in the process of liquid bridge formation and criteria based on static energy minimization approach has limited applicability.

This paper is the second part of a two part sequence on multiphysics algorithms and software. The first [1] focused on the algorithms; this part treats the multiphysics software framework and applications based on it. Tight coupling is typically designed into the analysis application at inception, as such an application is strongly tied to a composite nonlinear solver that arrives at the final solution by treating all equations simultaneously. The application must also take care to minimize both time and space error between the physics, particularly if more than one mesh representation is needed in the solution process. This paper presents an application framework that was specifically designed to support tightly coupled multiphysics analysis. The Multiphysics Object Oriented Simulation Environment (MOOSE) is based on the Jacobian-free Newton-Krylov (JFNK) method combined with physics-based preconditioning to provide the underlying mathematical structure for applications. The report concludes with the presentation of a host of nuclear, energy and environmental applications that demonstrate the efficacy of the approach and the utility of a well-designed multiphysics framework.

Optimal Replacement Variables (ORV) is a method for approximating a large system of ODEs by one with fewer equations, while attempting to preserve the essential dynamics of a reduced set of variables of interest. An earlier version of ORV [1] had some issues, including limited accuracy and in some rare cases, instability. Here we present a new version of ORV, inspired by the linear quadratic regulator problem of control theory, which provides better accuracy, a guarantee of stability and is in some ways easier to use.

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.

We propose a direct solver for the three-dimensional Poisson equation with a variable coefficient, and an algorithm to directly solve the associated sparse linear systems that exploits the sparsity pattern of the coefficient matrix. Introducing some appropriate finite difference operators, we derive a second-order scheme for the solver, and then two suitable high-order compact schemes are also discussed. For a cube containing N nodes, the solver requires arithmetic operations and memory to store the necessary information. Its efficiency is illustrated with examples, and the numerical results are analysed.

Solving elasticity equations with interfaces is a challenging problem for most existing methods. Nonetheless, it has wide applications in engineering and science. An accurate and efficient method is desired. In this paper, an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve elasticity equations with interfaces. The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface. The resulting linear system of equations is shown to be positive definite under certain assumptions. Numerical experiments show that this method is second order accurate in the L∞ norm for piecewise smooth solutions. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up) on the sharp-edged interface corner.

A two-dimensional numerical scheme for the compressible Euler equations is presented and applied here to the simulation of exemplary compressible vortical flows. The proposed approach allows to perform computations on unstructured moving grids with adaptation, which is required to capture complex features of the flow-field. Grid adaptation is driven by suitable error indicators based on the Mach number and by element-quality constraints as well. At the new time level, the computational grid is obtained by a suitable combination of grid smoothing, edge-swapping, grid refinement and de-refinement. The grid modifications—including topology modification due to edge-swapping or the insertion/deletion of a new grid node—are interpreted at the flow solver level as continuous (in time) deformations of suitably-defined node-centered finite volumes. The solution over the new grid is obtained without explicitly resorting to interpolation techniques, since the definition of suitable interface velocities allows one to determine the new solution by simple integration of the Arbitrary Lagrangian-Eulerian formulation of the flow equations. Numerical simulations of the steady oblique-shock problem, of the steady transonic flow and of the start-up unsteady flow around the NACA 0012 airfoil are presented to assess the scheme capabilities to describe these flows accurately.

We construct an efficient numerical scheme for the quantum Fokker-Planck- Landau (FPL) equation that works uniformly from kinetic to fluid regimes. Such a scheme inevitably needs an implicit discretization of the nonlinear collision operator, which is difficult to invert. Inspired by work [9] we seek a linear operator to penalize the quantum FPL collision term QqFPL in order to remove the stiffness induced by the small Knudsen number. However, there is no suitable simple quantum operator serving the purpose and for this kind of operators one has to solve the complicated quantum Maxwellians (Bose-Einstein or Fermi-Dirac distribution). In this paper, we propose to penalize QqFPL by the "classical" linear Fokker-Planck operator. It is based on the observation that the classical Maxwellian, with the temperature replaced by the internal energy, has the same first five moments as the quantum Maxwellian. Numerical results for Bose and Fermi gases are presented to illustrate the efficiency of the scheme in both fluid and kinetic regimes.

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 piece-wise polynomial are computed locally in a cell by the traditional DG method (called as “dynamic reconstruction”), while the higher-order derivatives are re-constructed 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.

In this paper, a newly developed second order temporally and spatially accurate finite difference scheme for biharmonic semi linear equations has been employed in simulating the time evolution of viscous flows past an impulsively started circular cylinder for Reynolds number (Re) up to 9,500. The robustness of the scheme and the effectiveness of the formulation can be gauged by the fact that it very accurately captures complex flow structures such as the von Kármán vortex street through streakline simulation and the α and β-phenomena in the range 3,000≤Re≤9,500 among others. The main focus here is the application of the technique which enables the use of the discretized version of a single semi linear biharmonic equation in order to efficiently simulate different fluid structures associated with flows around a bluff body. We compare our results, both qualitatively and quantitatively, with established numerical and more so with experimental results. Excellent comparison is obtained in all the cases.