A hybrid finite difference level set–implicit mesh discontinuous Galerkin method for multi-layer coating flows
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97 MATHEMATICS AND COMPUTING↗
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Implicit solvers for unstructured meshes
Implicit methods for unstructured mesh computations are developed and tested. The approximate system which arises from the Newton-linearization of the nonlinear evolution operator is solved by using the preconditioned generalized minimum residual technique. These different preconditioners are investigated: the incomplete LU factorization (ILU), block diagonal factorization, and the symmetric successive over-relaxation (SSOR). The preconditioners have been optimized to have good vectorization properties. The various methods are compared over a wide range of problems. Ordering of the unknowns, which affects the convergence of these sparse matrix iterative methods, is also investigated. Results are presented for inviscid and turbulent viscous calculations on single and multielement airfoil configurations using globally and adaptively generated meshes.
Venkatakrishnan, V.↗
Implicit solvers for unstructured meshes
Implicit methods were developed and tested for unstructured mesh computations. The approximate system which arises from the Newton linearization of the nonlinear evolution operator is solved by using the preconditioned GMRES (Generalized Minimum Residual) technique. Three different preconditioners were studied, namely, the incomplete LU factorization (ILU), block diagonal factorization, and the symmetric successive over relaxation (SSOR). The preconditioners were optimized to have good vectorization properties. SSOR and ILU were also studied as iterative schemes. The various methods are compared over a wide range of problems. Ordering of the unknowns, which affects the convergence of these sparse matrix iterative methods, is also studied. Results are presented for inviscid and turbulent viscous calculations on single and multielement airfoil configurations using globally and adaptively generated meshes.
Venkatakrishnan, V.↗
Implementation of Implicit Adaptive Mesh Refinement in an Unstructured Finite-Volume Flow Solver
This paper explores the implementation of adaptive mesh refinement in an unstructured, finite-volume solver. Unsteady and steady problems are considered. The effect on the recovery of high-order numerics is explored and the results are favorable. Important to this work is the ability to provide a path for efficient, implicit time advancement. A method using a simple refinement sensor based on undivided differences is discussed and applied to a practical problem: a shock-shock interaction on a hypersonic, inviscid double-wedge. Cases are compared to uniform grids without the use of adapted meshes in order to assess error and computational expense. Discussion of difficulties, advances, and future work prepare this method for additional research. The potential for this method in more complicated flows is described.
Schwing, Alan M.↗
Space Needle Returns
STEP is imported into Engineering Sketch Pad. Some bodies where slightly scaled and translated in OpenCSM to create a manifold solid for the downstream meshing process. The braces at the base and columns around the core are omitted because their solids are malformed or created nonmanifold intersections. EGADS provides an initial tessellation of the surface. refine adapted the surface mesh to a curvature and feature size metric. TetGen initially filled the volume. The TetGen mesh is adapted to the Spalding Law of the Wall u+ with refine to provide the initial mesh for flow solution. Solution-based mesh adaptation is performed where FUN3D-FV computes the flow solution with the Reynolds-averaged Navier-Stokes equations coupled to the Spalart-Allmaras turbulence model. The freestream Mach number is 4 approaching 40° from the central axis of the Space Needle. The volume and surface mesh is adapted with refine to reduce estimated interpolation error in Mach number via the multiscale metric. The adapted mesh implicitly resolves the boundary layers, shocks, and expansions. The surface mesh is shown for the lee side with a slice through the volume on the left. Computational schlieren in the lower right shows density variations. A slice of the mesh is colored with Mach number in the upper right where mesh with freestream Mach number is not rendered. The volume mesh contains 64 million vertices. A NASA worm logo is sketched and extruded into a solid in OpenCSM. The worm is unioned to the Space Needle roof to produce the inset mesh image.
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A fast implicit upwind solution algorithm for three-dimensional unstructured dynamic meshes
A fast implicit upwind algorithm for the solution of the time-dependent Euler equations is presented for aerodynamic analysis involving unstructured dynamic meshes. The spatial discretization of the scheme is based on the upwind approach of Roe, referred to as flux-difference splitting (FDS). The FDS approach is naturally dissipative and captures shock waves and contact discontinuities sharply. The temporal discretization of the scheme involves an implicit time-integration using a two-sweep Gauss-Seidel relaxation procedure. The procedure is computationally efficient for either steady or unsteady flow problems. A detailed description is given of the implicit upwind solution algorithm along with results which assess the capability. The results are presented for the NACA 0012 airfoil and for the Boeing 747 aircraft. The 747 geometry includes the fuselage, wing, horizontal and vertical tails, under-wing pylons, and flow-through engine nacelles. Euler solutions for the 747 aircraft on an unstructured tetrahedral mesh containing approximately 100,000 cells were obtained to engineering accuracy in less than one hour CPU time on a Cray-2 computer.
Batina, John T.↗
A fast implicit upwind solution algorithm for three-dimensional unstructured dynamic meshes
A fast implicit upwind algorithm for the solution of the time-dependent Euler equations is presented for aerodynamic analysis involving unstructured dynamic meshes. The spatial discretization of the scheme is based on the upwind approach of Roe, referred to as flux-difference splitting (FDS). The FDS approach is naturally dissipative and captures shock waves and contact discontinuities sharply. The temporal discretization of the scheme involves an implicit time-integration using a two-sweep Gauss-Seidel relaxation procedure. The procedure is computationally efficient for either steady or unsteady flow problems. A detailed description is given of the implicit upwind solution algorithm along with results which assess the capability. The results are presented for the NACA 0012 airfoil and for the Boeing 747 aircraft. The 747 geometry includes the fuselage, wing, horizontal and vertical tails, under-wing pylons, and flow-through engine nacelles. Euler solutions for the 747 aircraft on an unstructured tetrahedral mesh containing approximately 100,000 cells were obtained to engineering accuracy in less than one hour CPU time on a Cray-2 computer.
Batina, John T.↗
Calculation of two-dimensional inlet flow fields in a supersonic free stream by an implicit marching code with nonorthogonal mesh generation: User's manual
An implicit, shock-capturing finite-difference code which is used to calculate two-dimensional inlet flow fields in a supersonic free stream is explained. The Euler equations are subjected to general nonorthogonal transformation and a body-fitted coordinate system is employed. The mathematical formulation of the problem is given along with the numerical algorithm. Initial and boundary conditions, numerical stability, program limitations, and accuracy is discussed. An overall program logic as well as instructions for program use and operation are also furnished.
Biringen, S. H.↗
Implicit upwind solution algorithms for three-dimensional unstructured meshes
The development of implicit upwind algorithms for the solution of the three-dimensional, time-dependent Euler equations on unstructured tetrahedral meshes is described. The implicit temporal discretization involves either a two-sweep Gauss-Seide relaxation procedure, a two-sweep Point-Jacobi relaxation procedure, or a single-sweep Point-Implicit procedure; the upwind spatial discretization is based on the flux-difference splitting of Roe. Detailed descriptions of the three implicit solution algorithms are given, and calculations for the Boeing 747 transport configuration are presented to demonstrate the algorithms. Advantages and disadvantages of the implicit algorithms are discussed. A steady-state solution for the 747 configuration, obtained at transonic flow conditions using a mesh of over 100,000 cells, required less than one hour of CPU time on a Cray-2 computer, thus demonstrating the speed and robustness of the general capability.
Batina, John T.↗
Scalable Implicit Solvers with Dynamic Mesh Adaptation for a Relativistic Drift-Kinetic Fokker–Planck–Boltzmann Model
In this work we consider a relativistic drift-kinetic model for runaway electrons along with a Fokker–Planck operator for small-angle Coulomb collisions, a radiation damping operator, and a secondary knock-on (Boltzmann) collision source. Here, we develop a new scalable fully implicit solver utilizing finite volume and conservative finite difference schemes and dynamic mesh adaptivity. A new data management framework in the PETSc library based on the p4est library is developed to enable simulations with dynamic adaptive mesh refinement (AMR), distributed memory parallelization, and dynamic load balancing of computational work. This framework and the runaway electron solver building on the framework are able to dynamically capture both bulk Maxwellian at the low-energy region and a runaway tail at the high-energy region. To effectively capture features via the AMR algorithm, a new AMR indicator prediction strategy is proposed that is performed alongside the implicit time evolution of the solution. This strategy is complemented by the introduction of computationally cheap feature-based AMR indicators that are analyzed theoretically. Numerical results quantify the advantages of the prediction strategy in better capturing features compared with nonpredictive strategies; and we demonstrate trade-offs regarding computational costs. The robustness with respect to model parameters, algorithmic scalability, and parallel scalability are demonstrated through several benchmark problems including manufactured solutions and solutions of different physics models. We focus on demonstrating the advantages of using implicit time stepping and AMR for runaway electron simulations.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
An implicit algorithm for the conservative transonic full potential equation using an arbitrary mesh
A new, implicit approximate factorization (AF) algorithm designed to solve the conservative full-potential equation for the transonic flow past arbitrary airfoils has been developed. The new algorithm uses an upwind bias of the density coefficient to provide stability in supersonic regions. This allows the simple two- and three-banded matrix form of the AF scheme to be retained over the entire flow field, even in regions of supersonic flow. A numerical transformation is used to establish an arbitrary body-fitted finite-difference mesh. Airfoil pressure distributions have been computed and are in good agreement with independent results.
Holst, T. L.↗
Implicit solvers for unstructured meshes
Explore the source record for details and available documents.
Venkatakrishnan, V.↗
Implementation of Adaptive Mesh Refinement in an Implicit Unstructured Finite-Volume Flow Solver
No abstract available
Schwing, Alan↗
High-Order Mesh hr-adaptivity for Surface Fitting to Implicit Geometries
We present an ℎ𝑟-adaptivity framework for morphing a given mesh to fit a target surface prescribed as the zero isocontour of a discrete function. In this framework, high-order meshing is posed as a variational minimization problem that depends on the mesh quality prescribed via the target matrix optimization paradigm (TMOP) and position of a subset of mesh nodes with respect to the target surface. The proposed formulation ensures that the variational problem is converged and mesh quality degradation near the surface is limited, even when the mesh topology is incompatible with the target surface. Additionally, a mesh subset-based approach and ℎ-refinement is introduced to efficiently increase fitting accuracy while reducing the computational cost of the mesh morphing problem. The ℎ𝑟-adaptivity technique extends to different element types in two- and three-dimensions, and can be used in existing finite element and spectral element frameworks to obtain high-order body-fitted meshes. Various numerical experiments demonstrate the robustness and accuracy of the fitting approach for problems of practical interest such as Lagrangian hydrodynamics and topology optimization.
97 MATHEMATICS AND COMPUTING↗
Geometry Mesher
This program implements tetrahedral meshing for implicitly-defined geometries, focusing on meshing unions of geometric primitives like spheres and cylinders. This enables users to quickly generate quality meshes of truss lattice and filament structures.
Watts, SethE [Lawrence Livermore National Laborato↗
Finite-difference fluid dynamics computer mathematical models for the design and interpretation of experiments for space flight
Numerical methods are used to design a spherical baroclinic flow model experiment of the large scale atmosphere flow for Spacelab. The dielectric simulation of radial gravity is only dominant in a low gravity environment. Computer codes are developed to study the processes at work in crystal growing systems which are also candidates for space flight. Crystalline materials rarely achieve their potential properties because of imperfections and component concentration variations. Thermosolutal convection in the liquid melt can be the cause of these imperfections. Such convection is suppressed in a low gravity environment. Two and three dimensional finite difference codes are being used for this work. Nonuniform meshes and implicit iterative methods are used. The iterative method for steady solutions is based on time stepping but has the options of different time steps for velocity and temperature and of a time step varying smoothly with position according to specified powers of the mesh spacings. This allows for more rapid convergence. The code being developed for the crystal growth studies allows for growth of the crystal as the solid-liquid interface. The moving interface is followed using finite differences; shape variations are permitted. For convenience in applying finite differences in the solid and liquid, a time dependent coordinate transformation is used to make this interface a coordinate surface.
Roberts, G. O.↗
High-Order Mesh Morphing for Boundary and Interface Fitting to Implicit Geometries
Here, we propose a method that morphs high-order meshes such that their boundaries and interfaces coincide/align with implicitly defined geometries. Our focus is particularly on the case when the target surface is prescribed as the zero isocontour of a smooth discrete function. Common examples of this scenario include using level set functions to represent material interfaces in multimaterial configurations, and evolving geometries in shape and topology optimization. The proposed method formulates the mesh optimization problem as a variational minimization of the sum of a chosen mesh-quality metric using the Target-Matrix Optimization Paradigm (TMOP) and a penalty term that weakly forces the selected faces of the mesh to align with the target surface. The distinct features of the method are use of a source mesh to represent the level set function with sufficient accuracy, and adaptive strategies for setting the penalization weight and selecting the faces of the mesh to be fit to the target isocontour of the level set field. We demonstrate that the proposed method is robust for generating boundary- and interface-fitted meshes for curvilinear domains using different element types in 2D and 3D.
97 MATHEMATICS AND COMPUTING↗
Time integration algorithms for the two-dimensional Euler equations on unstructured meshes
Explicit and implicit time integration algorithms for the two-dimensional Euler equations on unstructured grids are presented. Both cell-centered and cell-vertex finite volume upwind schemes utilizing Roe's approximate Riemann solver are developed. For the cell-vertex scheme, a four-stage Runge-Kutta time integration, a fourstage Runge-Kutta time integration with implicit residual averaging, a point Jacobi method, a symmetric point Gauss-Seidel method and two methods utilizing preconditioned sparse matrix solvers are presented. For the cell-centered scheme, a Runge-Kutta scheme, an implicit tridiagonal relaxation scheme modeled after line Gauss-Seidel, a fully implicit lower-upper (LU) decomposition, and a hybrid scheme utilizing both Runge-Kutta and LU methods are presented. A reverse Cuthill-McKee renumbering scheme is employed for the direct solver to decrease CPU time by reducing the fill of the Jacobian matrix. A comparison of the various time integration schemes is made for both first-order and higher order accurate solutions using several mesh sizes, higher order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The results obtained for a transonic flow over a circular arc suggest that the preconditioned sparse matrix solvers perform better than the other methods as the number of elements in the mesh increases.
Slack, David C.↗