Eulerian method for multiphase interactions of soft solid bodies in fluids

We introduce an Eulerian approach for problems involving one or more soft solids immersed in a fluid, which permits mechanical interactions between all phases. The reference map variable is exploited to simulate finite-deformation constitutive relations in the solid(s) on the same fixed grid as the fluid phase, which greatly simplifies the coupling between phases. Our coupling procedure, a key contribution in the current work, is shown to be computationally faster and more stable than an earlier approach, and admits the ability to simulate both fluid--solid and solid--solid interaction between submerged bodies. The interface treatment is demonstrated with multiple examples involving a weakly compressible Navier--Stokes fluid interacting with a neo-Hookean solid, and we verify the method's convergence. The solid contact method, which exploits distance-measures already existing on the grid, is demonstrated with two examples. A new, general routine for cross-interface extrapolation is introduced and used as part of the new interfacial treatment

On Solving the Poisson Equation with Discontinuities on Irregular Interfaces: GFM and VIM

Publisher's version (útgefin grein)We analyze the accuracy of two numerical methods for the variable coefficient Poisson equation with discontinuities at an irregular interface. Solving the Poisson equation with discontinuities at an irregular interface is an essential part of solving many physical phenomena such as multiphase flows with and without phase change, in heat transfer, in electrokinetics, and in the modeling of biomolecules’ electrostatics. The first method, considered for the problem, is the widely known Ghost-Fluid Method (GFM) and the second method is the recently introduced Voronoi Interface Method (VIM). The VIM method uses Voronoi partitions near the interface to construct local configurations that enable the use of the Ghost-Fluid philosophy in one dimension. Both methods lead to symmetric positive definite linear systems. The Ghost-Fluid Method is generally first-order accurate, except in the case of both a constant discontinuity in the solution and a constant diffusion coefficient, while the Voronoi Interface Method is second-order accurate in the -norm. Therefore, the Voronoi Interface Method generally outweighs the Ghost-Fluid Method except in special case of both a constant discontinuity in the solution and a constant diffusion coefficient, where the Ghost-Fluid Method performs better than the Voronoi Interface Method. The paper includes numerical examples displaying this fact clearly and its findings can be used to determine which approach to choose based on the properties of the real life problem in hand.The research of Á. Helgadóttir was supported by the University of Iceland Research Fund 2015 under HI14090070. The researches of A. Guittet and F. Gibou were supported in part by the NSF under DMS-1412695 and DMREF-1534264.Peer Reviewe

Hybrid mesh/particle meshless method for modeling geological flows with discontinuous transport properties

In the present paper, we introduce the Finite Difference Method-Meshless Method (FDM-MM) in the context of geodynamical simulations. The proposed numerical scheme relies on the well-established FD method along with the newly developed “meshless” method and, is considered as a hybrid Eulerian/Lagrangian scheme. Mass, momentum, and energy equations are solved using an FDM method, while material properties are distributed over a set of markers (particles), which represent the spatial domain, with the solution interpolated back to the Eulerian grid. The proposed scheme is capable of solving flow equations (Stokes flow) in uniform geometries with particles, “sprinkled” in the spatial domain and is used to solve convection- diffusion problems avoiding the oscillation produced in the Eulerian approach. The resulting algebraic linear systems were solved using direct solvers. Our hybrid approach can capture sharp variations of stresses and thermal gradients in problems with a strongly variable viscosity and thermal conductivity as demonstrated through various benchmarking test cases. The present hybrid approach allows for the accurate calculation of fine thermal structures, offering local type adaptivity through the flexibility of the particle method

Imposing mixed Dirichlet–Neumann–Robin boundary conditions in a level-set framework

Pre-print (óritrýnt handrit)We consider the Poisson equation with mixed Dirichlet, Neumann and Robin boundary conditions on irregular domains. We describe a straightforward and efficient approach for imposing the mixed boundary conditions using a hybrid finite-volume/finite-difference approach, leveraging on the work of Gibou et al. (2002) [14], Ng et al. (2009) [30] and Papac et al. (2010) [33]. We utilize three different level set functions to represent the irregular boundary at which each of the three different boundary conditions must be imposed; as a consequence, this approach can be applied to moving boundaries. The method is straightforward to implement, produces a symmetric positive definite linear system and second-order accurate solutions in the L-infinity-norm in two and three spatial dimensions. Numerical examples illustrate the second-order accuracy and the robustness of the method. (C) 2015 Elsevier Ltd. All rights reserved.The research of Á. Helgadóttir, Y.T. Ng and F. Gibou were supported in part by ONR under grant agreement N00014-11-1-0027, by the National Science Foundation under grant agreement CHE 1027817 and by the W.M. Keck Foundation. The research of C. Min was supported in part by the Kyung Hee University Research Fund (KHU-20070608) in 2007 and by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2008-331-C00045)

A numerical method for solving the elliptic and elasticity interface problems

Interface problems arise when dealing with physical problems composed of different materials or of the same material at different states. Because of the irregularity along interfaces, many common numerical methods do not work, or work poorly, for interface problems. Matrix-coefficient elliptic and elasticity equations with oscillatory solutions and sharp-edged interfaces are especially complicated and challenging for most existing methods. An accurate and efficient method is desired. In 1999, the boundary condition capturing method was proposed to deal with Poisson equations with interfaces whose variable coefficients and solutions may be discontinuous. In 2003, a weak formulation was derived. Built on previous work that solves elliptic interface problems with two domains in two dimensions, this dissertation improves the accuracy in the presence of sharp-edged interfaces and extends to elasticity interface problems with two domains in two dimensions, elliptic interface problems with three domains in two dimensions, and elliptic interface problems with two domains in three dimensions. The method used in this dissertation is a non-traditional finite element method. The test function basis is chosen to be the standard finite element basis independent of the interface, and the solution basis is chosen to be piecewise linear, satisfying the jump conditions across the interface. These two bases are different, which leads to the non-symmetric matrix generated by this method, but the resulting linear system of equations is shown to be positive definite under certain assumptions in all the four topics mentioned in this dissertation. This method has matrix coefficients and lower-order terms, and uses the non-body-fitting grid which makes it easy to deal with different kinds of interfaces, like the examples “Star”, “Happy face”, “Chess board”, to name a few. The methods used in this dissertation solve the non-smooth interface case and promise results for oscillatory solutions. Numerical experiments show that this method is second-order accurate in the L ∞ norm for piecewise smooth solutions

An Implicit Interface Boundary Integral Method for Poisson’s Equation on Arbitrary Domains

We propose a simple formulation for constructing boundary integral methods to solve Poisson’s equation on domains with smooth boundaries defined through their signed distance function. Our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain, where the integrations are carried out in the level set framework using an appropriate Jacobian. By the coarea formula, the algorithm operates in the Euclidean space and does not require any explicit parameterization of the boundaries. We present numerical results in two and three dimensions