Implementation of the Ghost Fluid Method for Free Surface Flows in Polyhedral Finite Volume Framework

We present a method for solving Poisson and heat equations with discon- tinuous coefficients in two- and three-dimensions. It uses a Cartesian cut-cell/embedded boundary method to represent the interface between materi- als, as described in Johansen & Colella (1998). Matching conditions across the interface are enforced using an approximation to fluxes at the boundary. Overall second order accuracy is achieved, as indicated by an array of tests using non-trivial interface geometries. Both the elliptic and heat solvers are shown to remain stable and efficient for material coefficient contrasts up to 106, thanks in part to the use of geometric multigrid. A test of accuracy when adaptive mesh refinement capabilities are utilized is also performed. An example problem relevant to nuclear reactor core simulation is presented, demonstrating the ability of the method to solve problems with realistic physical parameters.

A Cartesian grid embedded boundary method for solving the Poisson and heat equations with discontinuous coefficients in three dimensions

A novel Douglas alternating direction implicit (ADI) method is proposed in this work to solve a two-dimensional (2D) heat equation with interfaces. The ADI scheme is a powerful finite difference method for solving parabolic equations, due to its unconditional stability and high efficiency. However, it suffers from a serious accuracy reduction in space for interface problems with different materials and nonsmooth solutions. If the jumps in a function and its derivatives are known across the interface, rigorous ADI schemes have been successfully constructed in the literature based on the immersed interface method so that the spatial accuracy can be restored. Nevertheless, the development of accurate and stable ADI methods for general parabolic interface problems with physical interface conditions that describe jumps of a function and its flux, remains unsolved. To overcome this difficulty, a novel tensor product decomposition is proposed in this paper to decouple 2D jump conditions into essentially one-dimensional (1D) ones. These 1D conditions can then be incorporated into the ADI central difference discretization, using the matched interface and boundary technique. Fast algebraic solvers for perturbed tridiagonal systems are developed to maintain the computational efficiency. Stability analysis is conducted through eigenvalue spectrum analysis, which numerically demonstrates the unconditional stability of the proposed ADI method. The matched ADI scheme achieves the first order of accuracy in time and second order of accuracy in space in all tested parabolic interface problems with complex geometries and spatial-temporal dependent jump conditions.

A Matched Alternating Direction Implicit (ADI) Method for Solving the Heat Equation with Interfaces

AP-Cloud

This study is aimed to analyze combined radiative and conductive heat transfer in two-dimensional irregular geometries using the embedded boundary treatment in Cartesian coordinate system. The main advantage of Cartesian formulation is to simplify grid generation and using efficient Cartesian solvers for problems with complex geometries. The discrete ordinates method is used for angular discretization of radiative transfer equation (RTE) in a participating medium and the finite volume method is used for spatial discretization of RTE and the energy equation. First, the required equations to implement the embedded boundary method in combined conductive-radiative problems are developed. Then, this method is employed to solve several coupled conductive-radiative problems associated with the complex geometry. To assess the accuracy and investigate various characteristics of the embedded boundary method, the results are compared with the other methods for analyzing of irregular geometries, i.e. the blocked-off method and the body-fitted treatment. The results showed that the embedded boundary method is an efficient approach for investigation heat transfer problems of combined conduction-radiation in complex enclosures including inclined or curved boundaries using Cartesian grid system. Especially for calculations near the complex boundaries, this method has high priority over the blocked-off method.

Coupled radiative-conductive heat transfer problems in complex geometries using embedded boundary method

An embedded boundary method for elliptic and parabolic problems with interfaces and application to multi-material systems with phase transitions

We develop an embedded boundary method (EBM) to solve the two-phase incompressible flow with piecewise constant density. The front tracking method is used to track the interface. The fractional step methods are used to solve the incompressible Navier-Stokes equations while the EBM is used in the projection step to solve an elliptic interface problem for the pressure with a jump equal to the surface tension force across the interface. Several examples are used to verify the accuracy of the method.

An Embedded Boundary Method for Two Phase Incompressible Flow

Transverse vibration simulation of sucker rod string under normally distributed load caused by viscoelastic fluid in a curved wellbore

In this paper, we perform a comparison study of two methods (the embedded boundary method and several versions of the mixed finite element method) to solve an elliptic boundary value problem.

Shuqiang Wang

Papers

An embedded boundary method for elliptic and parabolic problems with interfaces and application to multi-material systems with phase transitions

An Embedded Boundary Method for Two Phase Incompressible Flow

Transverse vibration simulation of sucker rod string under normally distributed load caused by viscoelastic fluid in a curved wellbore

A Comparison Study of Two Methods for Elliptic Boundary Value Problems