1,402 research outputs found
Grid generation for the solution of partial differential equations
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given
hp-FEM for Two-component Flows with Applications in Optofluidics
This thesis is concerned with the application of hp-adaptive finite element methods to a mathematical model of immiscible two-component flows. With the aim of simulating the flow processes in microfluidic optical devices based on liquid-liquid interfaces, we couple the time-dependent incompressible Navier-Stokes equations with a level set method to describe the flow of the fluids and the evolution of the interface between them
Mesh update techniques for free-surface flow solvers using spectral element method
This paper presents a novel mesh-update technique for unsteady free-surface
Newtonian flows using spectral element method and relying on the arbitrary
Lagrangian--Eulerian kinematic description for moving the grid. Selected
results showing compatibility of this mesh-update technique with spectral
element method are given
Mini-Workshop: Interface Problems in Computational Fluid Dynamics
Multiple difficulties are encountered when designing algorithms to simulate flows having free surfaces, embedded particles, or elastic containers. One difficulty common to all of these problems is that the associated interfaces are Lagrangian in character, while the fluid equations are naturally posed in the Eulerian frame. This workshop explores different approaches and algorithms developed to resolve these issues
A Cartesian grid-based boundary integral method for moving interface problems
This paper proposes a Cartesian grid-based boundary integral method for
efficiently and stably solving two representative moving interface problems,
the Hele-Shaw flow and the Stefan problem. Elliptic and parabolic partial
differential equations (PDEs) are reformulated into boundary integral equations
and are then solved with the matrix-free generalized minimal residual (GMRES)
method. The evaluation of boundary integrals is performed by solving equivalent
and simple interface problems with finite difference methods, allowing the use
of fast PDE solvers, such as fast Fourier transform (FFT) and geometric
multigrid methods. The interface curve is evolved utilizing the
variables instead of the more commonly used variables. This choice
simplifies the preservation of mesh quality during the interface evolution. In
addition, the approach enables the design of efficient and stable
time-stepping schemes to remove the stiffness that arises from the curvature
term. Ample numerical examples, including simulations of complex viscous
fingering and dendritic solidification problems, are presented to showcase the
capability of the proposed method to handle challenging moving interface
problems
Solving elliptic problems with discontinuities on irregular domains – the Voronoi Interface Method.
We introduce a simple method, dubbed the Voronoi Interface Method, to solve Elliptic problems with discontinuities across the interface of irregular domains. This method produces a linear system that is symmetric positive definite with only its right-hand-side affected by the jump conditions. The solution and the solution's gradients are second-order accurate and first-order accurate, respectively, in the L∞L∞ norm, even in the case of large ratios in the diffusion coefficient. This approach is also applicable to arbitrary meshes. Additional degrees of freedom are placed close to the interface and a Voronoi partition centered at each of these points is used to discretize the equations in a finite volume approach. Both the locations of the additional degrees of freedom and their Voronoi discretizations are straightforward in two and three spatial dimensions
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