Considerations on the spring analogy

• Published in: International journal for numerical methods in fluids 2000-03, Vol.32 (6), p.647-668
• Main Author: Blom, Frederic J.
• Format: Article
• Language: English
• Subjects: Aerodynamics; Applied fluid mechanics; Aspect ratio; Boundary conditions; Computational methods in fluid dynamics; Deformation; elliptic grid generation; Exact sciences and technology; Fluid dynamics; Functions; Fundamental areas of phenomenology (including applications); Inverse problems; Iterative methods; Mathematical models; mesh deformation; moving boundaries; Navier Stokes equations; Physics; Springs (components); Stiffness; unstructured meshes
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/(SICI)1097-0363(20000330)32:6<647::AID-FLD979>3.0.CO;2-K

This paper presents an investigation on the spring analogy. The spring analogy serves for deformation in a moving boundary problem. First, two different kinds of springs are discussed: the vertex springs and the segment springs. The vertex spring analogy is originally used for smoothing a mesh after mesh generation or refinement. The segment spring analogy is used for deformation of the mesh in a moving boundary problem. The difference between the two methods lies in the equilibrium length of the springs. By means of an analogy to molecular theory, the two theories are generalized into a single theory that covers both. The usual choice of the stiffness of the spring is clarified by the mathematical analysis of a representative one‐dimensional configuration. The analysis shows that node collision is prevented when the stiffness is chosen as the inverse of the segment length. The observed similarity between elliptic grid generation and the spring analogy is also investigated. This investigation shows that both methods update the grid point position by a weighted average of the surrounding points in an iterative manner. The weighting functions enforce regularity of the mesh. Based on these considerations, several improvements on the spring analogy are developed. The principle of Saint Venant is circumvented by a boundary correction. The prevention of inversion of triangular elements is improved by semi‐torsional springs. The numerical results show the superiority of the segment spring analogy over the vertex one for a small rotation of an NACA0012 mesh. The boundary correction allows for large deformation of the mesh, where the standard spring analogy fails. The final test is performed on a Navier–Stokes mesh. This mesh contains high aspect ratio mesh cells near the boundary. Large deformation of this mesh shows that the semi‐torsional spring improvement is imperative to retain the validity of this mesh. Copyright © 2000 John Wiley & Sons, Ltd.