# A Localized Meshless Technique for Generating 3-D Wind Fields

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mass Consistent Winds

_{0}, v

_{0}, and w

_{0}obtained from meteorological tower or SODAR-SOnic Detection and Ranging-data), along with Gauss moduli (α) that can be tuned to adjust for more horizontal or vertical effects (e.g., rough terrain may create more vertical influence). The resulting Euler-Lagrange equation for $\lambda \left(x,y,z\right)$ written as

_{i}are the Gauss precision moduli, where α

_{i}

^{2}≡ 1/(2σ

_{i}

^{2}) (with the deviation errors from the observed and desired fields defined by σ

_{i}). Sherman [4] points out that these moduli are important in establishing non-divergent wind fields over irregular terrain, where (α

_{1}/α

_{2})

^{2}is proportional to (w/u)

^{2}. Pepper and Wang [8] set α

_{1}(the horizontal adjustment) = 0.01 and α

_{2}(the vertical adjustment) = 0.1.

## 3. Wind Power Density Calculation

^{2}, wind velocity is m/s, and the density for air is 1.225 kg/m

^{3}at sea level. To account for density variation at elevation Z (above sea level in m), density is obtained using

## 4. The Meshless Method

#### 4.1. Radial Basis Functions (RBF)

_{j}is the Lagrange coefficient defined at each point.

**x**), a linear operator ($L\equiv {\nabla}^{2}$) is applied to the interior domain, Ω. Thus,

**x ≡**(x,y,z) with

**x**) denotes the divergence of the observed velocity values at the boundaries, Г. Introducing the MQ form of the basis function for ${\varphi}_{j}(x)$,

_{j}.

_{i}is the distance between the i th data point and its nearest neighbor. The shape parameter depends on the number and distribution of nodes, the choice of basis function, and computer precision [22].

#### 4.2. Local RBF Approach

**x**) of Equation (16), a series of local subdomains are solved that overlap within the problem domain. Figure 3 shows a set of subdomains with the dark points serving as the center node points. Each node serves as a center node of interest until all the nodes are resolved. This permits λ(

**x**) to be solved at every point, i.e.,

_{k,j}are the coefficients of the RBFs. The RBF, ${\varphi}_{k}$, are the shape functions. Substituting Equation (23) into Equation (21), an m x m linear algebraic system is obtained for each local domain with an interior point, i.e.,

## 5. Comparison Results

## 6. Implementation of the Meshless Method for Mobile Applications

## 7. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- Pepper, D.W. Utilization of Wind Energy in Nevada, Aerospace Sciences Meeting and Exhibit. In Proceedings of the 36th and 1998 ASME Wind Energy Symposium, Reno, NV, USA, 12–15 January 1998. [Google Scholar]
- AWS, Truewind, 2006. Available online: https://www.awstruepower.com/products/maps-and-resource-data/ (accessed on 3 February 2018).
- Cormier, C.K. “Energy for Nevada”, Report to the Legislature on the Status of Energy in Nevada for the Year 1996; Department of Business and Industry, Nevada, State Energy Office: Carson City, NV, USA, 1996; p. 36.
- Sherman, C.A. A Mass-consistent model for wind field over complex terrain. J. Appl. Meteor.
**1978**, 17, 312–319. [Google Scholar] [CrossRef] - Pepper, D.W. 3-D Numerical Model for Predicting Mesoscale Wind Fields over Vandenberg Air Force Base, Final Report; SBIR Contract F04701-89-C-0051; Advanced Project Research Inc.: Moorpark, CA, USA, 1990. [Google Scholar]
- Pielke, R.A. Mesoscale Meteorological Modeling; Academic Press: New York, NY, USA, 1984; p. 612. [Google Scholar]
- Sasaki, Y. An objective analysis based on the variational method. J. Meteor. Soc. Jpn.
**1958**, 36, 77–88. [Google Scholar] [CrossRef] - Pepper, D.W.; Wang, X. An h-Adaptive Finite-Element Technique for Constructing 3-D Wind Fields. J. Appl. Meteor. Clim.
**2009**, 48, 580–599. [Google Scholar] [CrossRef] - Dickerson, M.H. MASCON-A mass consistent atmospheric flux model for regions with complex terrain. J. Appl. Meteor.
**1978**, 17, 241–253. [Google Scholar] [CrossRef] - Renewable Resource Data Center: Wind Energy Resource Information-Wind Energy Resource Atlas of the United States, Chapter 1 Introduction-Map Descriptions. Available online: http://rredc.nrel.gov/wind/pubs/atlas/tables/1-1T.html (accessed on 3 February 2018).
- Fasshauer, G.E. Newton iteration with multiquadrics for the solution of nonlinear pdes. Comput. Math. Appl.
**2002**, 43, 423–438. [Google Scholar] [CrossRef] - Fasshauer, G.E. Meshfree Approximation Methods with MATLAB. In Interdisciplinary Mathematical Sciences; World Scientific: Singapore, 2007; Volume 6, p. 500. [Google Scholar]
- Atluri, S.N.; Zhu, T. A New Mesh-less Local Petrov-Galerkin Approach in Computational Mechanics. Comput. Mech.
**1998**, 22, 117–127. [Google Scholar] [CrossRef] - Balachandran, G.R.; Rajagopal, A.; Sivakumar, S.M. Mesh free Galerkin Method Based on Natural Neighbors and Conformal Mapping. Comput. Mech.
**2009**, 42, 885–905. [Google Scholar] [CrossRef] - Liu, G.R. Mesh Free Methods, Moving Beyond the Finite Element Method; CRC Press: Boca Raton, FL, USA, 2003; p. 692. [Google Scholar]
- Li, H.; Mulay, S.S. Meshless Methods and Their Numerical Properties; CRC Press: Boca Raton, FL, USA, 2013; p. 429. [Google Scholar]
- Choi, Y.; Kim, S.J. Node Generation Scheme for the Mesh-less Method by Voronoi Diagram and Weighted Bubble Packing. In Proceedings of the Fifth US National Congress on Computational Mechanics, Boulder, CO, USA, 4–6 August 1999. [Google Scholar]
- Gewali, L.; Pepper, D.W. Adaptive Node Placement for Mesh-Free Methods. In Proceedings of the International Conference on Computational & Experimental Engineering and Sciences 2010 (ICCES’10), Las Vegas, NV, USA, 28 March–1 April 2010. [Google Scholar]
- Franke, R. Scattered data interpolation tests of some methods. Math Comput.
**1982**, 38, 181–200. [Google Scholar] - Hardy, R.L. Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res.
**1971**, 76, 1905–1915. [Google Scholar] [CrossRef] - Kansa, E.J. Highly accurate methods for solving elliptic and parabolic partial differential equations. WIT Trans. Model. Simul.
**2005**, 39, 5–15. [Google Scholar] - Roque, C.M.C.; Ferreira, A.J.M. Numerical Experiments on Optimal Shape Parameters for Radial Basis Functions. Numer. Methods. Partial Differ. Equ.
**2009**, 26, 675–689. [Google Scholar] [CrossRef] - Pepper, D.W.; Rasmussen, C.; Fyda, D. A Meshless Method for Creating 3-D Wind Fields using Sparse Meteorological Data. CAMES
**2014**, 21, 233–243. [Google Scholar] - Waters, J.; Pepper, D.W. Global versus Localized RBF Meshless Methods for Solving Incompressible Fluid Flow with Heat Transfer. Numer. Heat Transf. B
**2015**, 68, 185–203. [Google Scholar] [CrossRef] - Pepper, D.W.; Wang, X.; Carrington, D.B. A Meshless Method for Modeling Convective Heat Transfer. ASME J. Heat Transf.
**2012**, 135. [Google Scholar] [CrossRef] - Pepper, D.W.; Wang, X. Application of an h-adaptive finite element model for wind energy assessment in Nevada. Renew. Energy
**2007**, 32, 1705–1722. [Google Scholar] [CrossRef] - Ramos Gonzalez, M.; Pepper, D.W. A Cloud-based Method for Displaying 3-D Wind Fields on Mobile Devices. In Proceedings of the AMS 33rd Conference on Environmental Information Processing Technologies, Seattle, WA, USA, 22–26 January 2017. [Google Scholar]

**Figure 1.**(

**a**) National Renewable Energy Laboratory and AWS Truepower—Nevada 50 m wind power map; (

**b**,

**c**) University of Nevada Las Vegas (UNLV) assessment (red-Class 7, Orange-Class 6, Yellow-Class 5, Green-Class 4)—1998 study.

**Figure 3.**Node placement and circle of influence (from Pepper, D.W., et al. [25]).

**Figure 4.**Localized stencil, (

**a**) 9-point and (

**b**) random 30-point array (from Waters, J., et al. [24]).

**Figure 5.**Steady-state conduction in a two-dimensional plate (from Waters, J., et al. [24]).

**Figure 8.**(

**a**) Local Meshless results for 10 m level and (

**b**) h-fem results. Figures indicate velocity vectors with tower locations indicated by the red markers on both figures (from Pepper, D.W., et al. [8]; ©American Meteorological Society, used with permission).

**Figure 9.**(

**a**) Local Meshless results for 50 m level and (

**b**) h-fem results. Figures indicate velocity vectors with tower locations indicated by the red markers on both figures (from Pepper, D.W., et al. [8]; ©American Meteorological Society, used with permission).

**Figure 13.**Power density contours for September 2001–February 2002. (

**a**) 50 m and (

**b**) 100 m (W/m

^{2}) (September 2001); (

**c**) 50 m and (

**d**) 100 m (W/m

^{2}) (October 2001); (

**e**) 50 m and (

**f**) 100 m (W/m

^{2}) (November 2001); (

**g**) 50 m and (

**h**) 100 m (W/m

^{2}) (December 2001); (

**i**) 50 m and (

**j**) 100 m (W/m

^{2}) (January 2002); (

**k**) 50 m and (

**l**) 100 m (W/m

^{2}) (February 2002).

Class | Power Density (W/m^{2}) | Mean Speed(m/s) |
---|---|---|

1 | <200 | <5.6 |

2 | 200–300 | 5.6–6.4 |

3 | 300–400 | 6.4–7.0 |

4 | 400–500 | 7.0–7.5 |

5 | 500–600 | 7.5–8.0 |

6 | 600–700 | 5.6–8.8 |

7 | >800 | >8.8 |

**Table 2.**Comparison of results for Exact, FEM, and Meshless Method (from Pepper, D.W., et al. [25]).

Method | Mid-Point (°C) | Elements | Nodes |
---|---|---|---|

Exact | 94.512 | 0 | 0 |

FEM | 94.605 | 256 | 289 |

Meshless | 94.514 | 0 | 325 |

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**MDPI and ACS Style**

Pepper, D.W.; Ramos Gonzalez, M.
A Localized Meshless Technique for Generating 3-D Wind Fields. *Computation* **2018**, *6*, 17.
https://doi.org/10.3390/computation6010017

**AMA Style**

Pepper DW, Ramos Gonzalez M.
A Localized Meshless Technique for Generating 3-D Wind Fields. *Computation*. 2018; 6(1):17.
https://doi.org/10.3390/computation6010017

**Chicago/Turabian Style**

Pepper, Darrell W., and Maria Ramos Gonzalez.
2018. "A Localized Meshless Technique for Generating 3-D Wind Fields" *Computation* 6, no. 1: 17.
https://doi.org/10.3390/computation6010017