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

Afivo: a framework for quadtree/octree AMR with shared-memory parallelization and geometric multigrid methods

Afivo is a framework for simulations with adaptive mesh refinement (AMR) on quadtree (2D) and octree (3D) grids. The framework comes with a geometric multigrid solver, shared-memory (OpenMP) parallelism and it supports output in Silo and VTK file formats. Afivo can be used to efficiently simulate AMR problems with up to about $10^{8}$ unknowns on desktops, workstations or single compute nodes. For larger problems, existing distributed-memory frameworks are better suited. The framework has no built-in functionality for specific physics applications, so users have to implement their own numerical methods. The included multigrid solver can be used to efficiently solve elliptic partial differential equations such as Poisson's equation. Afivo's design was kept simple, which in combination with the shared-memory parallelism facilitates modification and experimentation with AMR algorithms. The framework was already used to perform 3D simulations of streamer discharges, which required tens of millions of cells

Block Structured Adaptive Mesh and Time Refinement for Hybrid, Hyperbolic + N-body Systems

We present a new numerical algorithm for the solution of coupled collisional and collisionless systems, based on the block structured adaptive mesh and time refinement strategy (AMR). We describe the issues associated with the discretization of the system equations and the synchronization of the numerical solution on the hierarchy of grid levels. We implement a code based on a higher order, conservative and directionally unsplit Godunov's method for hydrodynamics; a symmetric, time centered modified symplectic scheme for collisionless component; and a multilevel, multigrid relaxation algorithm for the elliptic equation coupling the two components. Numerical results that illustrate the accuracy of the code and the relative merit of various implemented schemes are also presented.Comment: 40 pages, 10 figures, JPC in press. Extended the code test section, new convergence tests, several typos corrected. Full resolution version available at http://www.exp-astro.phys.ethz.ch/miniati/charm.pd

Large-Eddy Simulations of Flow and Heat Transfer in Complex Three-Dimensional Multilouvered Fins

The paper describes the computational procedure and results from large-eddy simulations in a complex three-dimensional louver geometry. The three-dimensionality in the louver geometry occurs along the height of the fin, where the angled louver transitions to the flat landing and joins with the tube surface. The transition region is characterized by a swept leading edge and decreasing flow area between louvers. Preliminary results show a high energy compact vortex jet forming in this region. The jet forms in the vicinity of the louver junction with the flat landing and is drawn under the louver in the transition region. Its interaction with the surface of the louver produces vorticity of the opposite sign, which aids in augmenting heat transfer on the louver surface. The top surface of the louver in the transition region experiences large velocities in the vicinity of the surface and exhibits higher heat transfer coefficients than the bottom surface.Air Conditioning and Refrigeration Project 9

Numerical and adaptive grid methods for ideal magnetohydrodynamics

In this thesis numerical finite difference methods for ideal magnetohydrodynamics (MHD) are investigated. A review of the relevant physics, essential for interpreting the results of numerical solutions and constructing validation cases, is presented. This review includes a discusion of the propagation of small amplitude waves in the MHD system as well as a thorough discussion of MHD shocks, contacts and rarefactions and how they can be pieced together to obtain a solutions to the MHD Riemann problem. Numerical issues relevant to the MHD system such as: the loss of nonlinear numerical stability in the presence of discontinuous solutions, the introduction of spurious forces due to the growth of the divergence of the magnetic flux density, the loss of pressure positivity, and the effects of non-conservative numerical methods are discussed, along with the practical approaches which can be used to remedy or minimize the negative consequences of each. The use of block structured adaptive mesh refinement is investigated in the context of a divergence free MHD code. A new method for conserving magnetic flux across AMR grid interfaces is developed and a detailed discussion of our implementation of this method using the CHOMBO AMR framework is given. A preliminary validation of the new method for conserving magnetic flux density across AMR grid interfaces illustrates that the method works. Finally a number of code validation cases are examined spurring a discussion of the strengths and weaknesses of the numerics employed

Advances in Time-Domain Electromagnetic Simulation Capabilities Through the Use of Overset Grids and Massively Parallel Computing

A new methodology is presented for conducting numerical simulations of electromagnetic scattering and wave propagation phenomena. Technologies from several scientific disciplines, including computational fluid dynamics, computational electromagnetics, and parallel computing, are uniquely combined to form a simulation capability that is both versatile and practical. In the process of creating this capability, work is accomplished to conduct the first study designed to quantify the effects of domain decomposition on the performance of a class of explicit hyperbolic partial differential equations solvers; to develop a new method of partitioning computational domains comprised of overset grids; and to provide the first detailed assessment of the applicability of overset grids to the field of computational electromagnetics. Furthermore, the first Finite Volume Time Domain (FVTD) algorithm capable of utilizing overset grids on massively parallel computing platforms is developed and implemented. Results are presented for a number of scattering and wave propagation simulations conducted using this algorithm, including two spheres in close proximity and a finned missile