The meshless (or mesh-free) method with RBFs is a relatively recent development that has been shown to successfully alleviate meshing problems common to finite difference, finite volume, and finite element methods – no mesh is required. There exist many types of meshless methods, such as kernel methods, moving least squares, partition of unity, and radial basis functions. The technique yields fast convergence and high accuracy necessary for providing quick forecasts. These methods have the following advantages: (i) they require neither domain nor boundary discretization; (ii) domain integration is not required; (iii) they converge exponentially for smooth boundary shape and boundary data; (iv) they are insensitive to the cause of dimension and thus attractive to high dimensional problems; and (v) implementation and coding are very easy.
In general, there are two major meshless developments: the first is the so-called MFS-DRM which evolved from the dual reciprocity boundary element method (DRBEM) and used RBFs for approximating the source term; the second mesh-free method using RBFs is Kansa's method (1990), where the RBFs are directly implemented for the approximation of the solution of partial differential equations (PDEs). Both techniques have been used to solve various classes of science and engineering problems; however, direct application to geophysical and environmental problems have been very limited [Goldberg and Chen, 1997].
The hp-adaptive finite element method employs h-adaptation for mesh refinement and p-adaptation for element basis functions (linear, quadratic, cubic, etc.). The combination of both creates a highly accurate computational method and solution strategy (basis functions) that changes in response to an evolving solution. Complex flow regions are often regions with high gradients and large numerical errors. The number of collocation points and/or interpolation order in regions of high gradients are increased (or refined), and the number of collocation points, or lower interpolation order, is implemented (unrefined) where the flow is smooth. An important advantage of using the h-p technique is that it increases resolution in regions of complex terrain. This particular technique permits ground features to be resolved at meter levels (or smaller), while still being able to accommodate regional mesoscale or sub-mesoscale flow phenomena. The end result is a numerical method that automatically refines and unrefines as the solution progresses, thereby increasing both the solution accuracy and speed of convergence. This double adaptive feature of the method, while yielding exponential convergence, requires care in book keeping of elements and handling of virtual nodes (where higher order elements are adjacent to lower order elements).
The DOD, along with several government sponsored National Laboratories, conducted extensive studies to model and experimentally verify airflow and diffusion in complex terrain (also see Goodin, et al, 1980). Efforts include the Air Force Space Division for Space Shuttle impact modeling, the Army Atmospheric Science Laboratory for validation of 3-D flow models, and the Naval Postgraduate School for parameterizing coastal diffusion including a complex terrain puff model. Numerous field measurements and numerical modeling of flows over Vandenberg Air Force Base (VAFB) in California have also been made over many years. The VAFB terrain is quite hilly, and many field experiments to assess puff/plume trajectories were conducted. Kamada, et al (1991) examined wind fields and eight plume releases from Mt. Iron. A 3-D, non-adapting finite element model was developed by Pepper (1990) for VAFB. In addition, there have been several large-scale measurement programs carried out in complex terrain, including the Atmospheric Studies in Complex Terrain (ASCOT), coordinated by the Lawrence Livermore National Laboratory for the Department of Energy [Dickerson and Gudiksen, 1980]. The purpose of ASCOT was to study nocturnal drainage flows, concentrating on intensive measurement periods of fairly short duration. More recently, a simulation of 3-D wind fields over the Nevada Test Site (NTS) was conducted by Pepper and Wang (2009) using an h-adaptive finite element model.
In this study, an hp-adaptive finite element technique is compared with a meshless method for constructing 3-D wind fields from sparse meteorological tower data, including dispersion from a point source. Using data obtained from the NTS, output from each model is examined and results compared. The hp-adaptive model is run on the Cherry Creek HPC maintained by the National Supercomputer Center for Energy and the Environment at UNLV; this supercomputer, built by Intel, contains 26,000 computational cores. The meshless method can be run on a PC. Both techniques yield accurate results.
1. Dickerson, M. H. and Gudiksen, P. H. (1980) ASCOT FY-1979 Progress Report, UCRL-52899/ascot/LLNL, Livermore, CA. 2. Golberg, M.A. and Chen, C.S. (1997): Discrete Projection Methods for Integral Equations, Computational Mechanics Publications, Southhampton. 3. Goodin, W. R., McRae, G. J., and Seinfeld, J. H. (1980) An Objective Analysis Technique for Constructing Three-Dimensional Urban Scale Wind Fields, J. Appl. Meteor., 19, pp. 98-108. 5. Kamada, R. F., Drake, S. A., Mikkelsen, T., and Thykier-Nielsen, S. (1991) NPS-PH-92-006, Naval Postgraduate School, Monterey, CA, 85 p. 6. Kansa, E.J. (1990): “Multiquatric – A scattered data approximation scheme with applications to computational fluid dynamics II”, Computers Math. Appl., Vol. 19, No. 8/9, pp.147-161. 7. Pepper, D. W. (1991) A Finite Element Model for Calculating 3-D Wind Fields over Vandenberg Air Force Base, 29th AIAA Aerospace Sciences Meeting, Jan. 7-10, Reno, NV, AIAA Paper 91-0451. 8. Pepper, D. W. and X. Wang (2009): An h-Adaptive Finite Element Technique for Constructing 3D Wind Fields, J. Appl. Meteor. and Climat., 48, pp. 580-599.