Boundary-fitted coordinate systems for numerical solution of partial differential equations—A review
TL;DR: A comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given in this article, along with a general mathematical framework and error analysis common to such coordinate systems.
About: This article is published in Journal of Computational Physics.The article was published on 1982-07-01. It has received 542 citations till now. The article focuses on the topics: Ellipsoidal coordinates & Spherical coordinate system.
Citations
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TL;DR: The basic explicit finite element and finite difference methods that are currently used to solve transient, large deformation problems in solid mechanics are reviewed.
Abstract: Explicit finite element and finite difference methods are used to solve a wide variety of transient problems in industry and academia. Unfortunately, explicit methods are rarely discussed in detail in finite element text books. This paper reviews the basic explicit finite element and finite difference methods that are currently used to solve transient, large deformation problems in solid mechanics. A special emphasis has been placed on documenting methods that have not been previously published in journals.
1,218 citations
01 Jan 1985
TL;DR: A three-dimensional (3-D) chimera grid-embedding technique that simplifies the construction of computational grids about complex geometries by solution of the Euler equations for the transonic flow about a wing/body, wing/ body/tail, and a configuration of three ellipsoidal bodies is described.
Abstract: A three-dimensional (3-D) chimera grid-embedding technique is described. The technique simplifies the construction of computational grids about complex geometries. The method subdivides the physical domain into regions which can accommodate easily generated grids. Communication among the grids is accomplished by interpolation of the dependent variables at grid boundaries. The procedures for constructing the composite mesh and the associated data structures are described. The method is demonstrated by solution of the Euler equations for the transonic flow about a wing/body, wing/body/tail, and a configuration of three ellipsoidal bodies.
615 citations
01 Jan 1983
TL;DR: An automated grid embedding procedure for solution of flows about complex geometries is described, and the method is demonstrated by solution of the Euler equations for transonic flow about a supercritical airfoil and a flapped airfoils.
Abstract: An automated grid embedding procedure for solution of flows about complex geometries is described. The physical domain is subdivided into regions that can accommodate easily generated grids. The grids are organized in a hierarchical structure, and communication among grids is accomplished by interpolation of the flow variables at mesh boundaries. Algorithms for locating embedded boundaries, special treatment of the embedded grids, and the data structures required for manipulating the solution data are described. The method is demonstrated by solution of the Euler equations for transonic flow about a supercritical airfoil and a flapped airfoil.
379 citations
TL;DR: In this paper, a cubic-polynomial interpolation method, where the gradient of the quantity is a free parameter, is proposed for solving hyperbolic-type equations, and various choices of the gradient are investigated, and a stable and less diffusive scheme is made possible without clipping or the flux-correction procedure.
338 citations
TL;DR: In this article, a simple Lagrangian-Eulerian formulation of finite element programs is presented, where an operator split separates the Lagrangians and Eulerian processes, allowing a finite element program to be extended to this formulation with little difficulty.
Abstract: A simple arbitrary Lagrangian-Eulerian formulation is presented. An operator split separates the Lagrangian and Eulerian processes, allowing a Lagrangian finite element program to be extended to this formulation with little difficulty. Example problems illustrate the strengths and weaknesses of the formulation.
305 citations
References
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TL;DR: In this paper, the boundary value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes, and interactions between these levels enable us to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); and conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining "°°-order" approximations and low n, even when singularities are present.
Abstract: The boundary-value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes. Interactions between these levels enable us (i) to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); (ii) to conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining \"°°-order\" approximations and low n, even when singularities are present. General theoretical analysis of the numerical process. Numerical experiments with linear and nonlinear, elliptic and mixed-type (transonic flow) problemsconfirm theoretical predictions. Similar techniques for initial-value problems are briefly
3,038 citations
TL;DR: In this article, a geometric conservation law (GCL) is formulated that governs the spatial volume element under an arbitrary mapping and the GCL is solved numerically along with the flow conservation laws using conservative difference operators.
Abstract: Boundary-conforming coordinate transformations are used widely to map a flow region onto a computational space in which a finite-difference solution to the differential flow conservation laws is carried out. This method entails difficulties with maintenance of global conservation and with computation of the local volume element under time-dependent mappings that result from boundary motion. To improve the method, a differential ''geometric conservation law" (GCL) is formulated that governs the spatial volume element under an arbitrary mapping. The GCL is solved numerically along with the flow conservation laws using conservative difference operators. Numerical results are presented for implicit solutions of the unsteady Navier-Stokes equations and for explicit solutions of the steady supersonic flow equations.
1,188 citations
TL;DR: In this paper, a method for automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of general multi-connected regions containing any number of arbitrarily shaped bodies is presented.
996 citations
TL;DR: In this article, an implicit finite-difference procedure for unsteady 3D flow capable of handling arbitrary geometry through the use of general coordinate transformations is described, where viscous effects are optionally incorporated with a "thin-layer" approximation of the Navier-Stokes equations.
Abstract: An implicit finite-difference procedure for unsteady three-dimensional flow capable of handling arbitrary geometry through the use of general coordinate transformations is described. Viscous effects are optionally incorporated with a "thin-layer" approximation of the Navier-Stokes equations. An implicit approximate factorization technique is employed so that the small grid sizes required for spatial accuracy and viscous resolution do not impose stringent stability limitations. Results obtained from the program include transonic inviscid or viscous solutions about simple body configurations. Comparisons with existing theories and experiments are made. Numerical accuracy and the effect of three-dimensional coordinate singularities are also discussed.
769 citations
TL;DR: In this paper, an automatic grid generation program is employed, and because an implicit finite-difference algorithm for the flow equations is used, time steps are not severely limited when grid points are finely distributed.
Abstract: Finite-difference procedures are used to solve either the Euler equations or the "thin-layer" Navier-Stokes equations subject to arbitrary boundary conditions. An automatic grid generation program is employed, and because an implicit finite-difference algorithm for the flow equations is used, time steps are not severely limited when grid points are finely distributed. Computational efficiency and compatibility to vectorized computer processors is maintained by use of approximate factorization techniques. Computed results for both inviscid and viscous flow about airfoils are described and compared to viscous known solutions.
691 citations