A Finite-Difference Logical-Coordinate Evaluation Method for Particle Localization

Abstract

Particle methods require robust and efficient advection and localization methods which include logical-coordinate evaluation. The ability to compute logical coordinates with existing methods is, however, not guaranteed within grids that contain nonlinear elements. This note presents a new logical-coordinate evaluation method, based on finite-differences, that provides a robust and efficient solution for coordinate transformation. The new methods enhanced capabilities are demonstrated on a simple test problem. §1. IntroductionParticle methods, computational models of particle dynamics, require robust and efficient advection and localization methods. Localization methods 1) -6) combine cell-searching and logical-coordinate evaluation methods to define particle-grid connectivity. This connectivity data consist of the identity of the grid cell in which the particle resides and the particle's position relative to that cell, its transformed or logical coordinates. Cell-searching or guessing methods typically use the particle's logical coordinates to both direct and halt the search. Particle methods are, therefore, predicated on robust and efficient logical-coordinate evaluation methods.Existing logical-coordinate evaluation methods are generalized in Ref. 6). Solutions using this technique are guaranteed, and the coordinate vector is bound between known transformation limits, if the particle resides within the guessed cell. In contrast, an unbound coordinate solution may fail to exist for nonlinear grid element transformations. The problem of interest, however, occurs when these coordinates are unbound because only then will the particle have exited the cell during advection. This note continues by presenting a new evaluation method that is less sensitive to coordinate transformation. A test problem concludes this note. §2. Finite-difference evaluation methodEvaluating logical coordinates is the transformation from physical, X = (x, y, z) T , to logical, ξ = (ξ, η, ζ) T , spatial coordinates. Around complex geometries, computational space is often discretized into nonorthogonal hexahedral cells. A trilinear function, X(ξ, X cv ), where ξ is bound if 0 ≤ξ≤ 1, is generally utilized in these cells for both data interpolation and coordinate transformation. The vector X cv defines the cell-vertex (cv) physical coordinates.The new logical-coordinate evaluation method is developed by considering the difference between two discrete particle locations, ∆X = X(ξ 2 ,X cv 2 ) − X(ξ 1 ,X cv 1 ).

Integrating a Bilinear Interpolation Function Across Quadrilateral Cell Boundaries

“…In contrast, the existing finitedifference method [73][74][75] produced only a single discrete-expansion of trilinear interpolation defined within hexahedral cells. Furthermore, the finite-difference expansion is a subset of the total-differential solutions; it is the three-dimensional equivalent of one of the total-differential expansions.…”

“…The second expression within Equation 22 is the two-dimensional equivalent of the discrete-expansion obtained using the finite-difference method [73][74][75]. Therefore, the new totaldifferential and the existing finite-difference methods of developing discrete-expansions produce identical results for similar computational cell geometries; bilinear interpolation defined within quadrilateral cells is a subset of the trilinear function defined within hexahedral cell geometries.…”

“…Furthermore, the linear discrete-expansions are presented here for completeness; discreteexpansions for bilinear interpolation are presented in this report, and they have been previously reported for trilinear interpolation using the finite-difference method [73][74][75].…”

“…This expansion is also the one-dimensional equivalent of the discrete-expansion obtained with the existing finite-difference method [73][74][75]. …”