Green’s function–based surrogate model for windfields using limited samples

Abstract

There exists many applications for which wind-velocity is desired over a three-dimensional space. The vector field associated with these wind velocities is known as a “windfield” or “velocity-windfield.” The present work provides a fast method to characterize windfields. The approach uses the free-space Green’s function for potential theory as an inexpensive surrogate model in lieu of either complicated physics-based models or other types of surrogate models, both of which require volumetric discretizations for the three-dimensional case. Using the gradient of the third Green’s identity, the wind-velocity in the interior of a domain is entirely characterized by a surface discretization while still providing a three-dimensional model. The unknown densities on the surface are determined from enforcement of the interior form of the identity at arbitrary points coinciding with wind measurements taken by unmanned aerial vehicles. Numerical results support the feasibility of the method.

Keywords

Method of interior equations windfield Green’s function potential theory interior conditions

Introduction

The present work introduces a “wind-mapper” simulation strategy developed to approximately represent local windfields over some range based on limited data acquired recently to the time of interest. In this study, data are anticipated to be in the form of air speed as measured by coordinated fleets of unmanned aerial vehicles (UAVs). Applications where mission success would depend heavily on wind conditions include activities such as aerial dispersal of chemical agents used in fighting forest fires (Sanjuan et al., 2014), and in agriculture (Physick et al., 1992), along with parachute drops of either supplies or personnel (Cambell et al., 2005). When the expense of failure to deliver to a given target is sufficiently high, it would be attractive to invest in a UAV fleet capable of measuring and transmitting wind data on a local level followed by a fast simulation which could characterize the windfield throughout some region based on limited data.

Several numerical methods have been proposed to characterize wind-velocity including radial basis functions (RBFs; Hickernell and Hon, 1998; Montalvo et al., 2015), the finite difference method (FDM; Hacène et al., 2012; Montero and Sanın, 2001), and the finite element method (FEM; Caneill et al., 1984; Huang et al., 2011; Montero et al., 1998; Pepper and Wang, 2009; Winter et al., 1995), among others. Each of the aforementioned methods are domain-based and as such the data structure of the models will scale as $O (L^{3})$ where $L$ is the characteristic length of the domain of interest. However, boundary-based methods (Becker, 1992) are well known to reduce a problem’s dimensionality by 1 in comparison with domain methods which is particularly attractive in terms of data collection.

Use of interior conditions in lieu of boundary conditions

The standard form for posing initial-boundary value problems is generally consistent with the temporal and physical availability of data. As an example, a standard boundary value problem of potential theory seeks to identify a potential function consistent with some specified conditions involving the potential and its normal derivatives on the boundary, specifically where the system obeying the governing partial differential equation interacts with its surroundings through boundary interaction. Boundary value problems are typically posed in this fashion since any measurement device cannot generally gain access to the interior of a solid object or a domain without either damaging the object in some way or modifying the field. A windfield is no exception. However, for large windfields, small UAVs may obtain interior measurements without significantly altering the wind itself.

With this idea, one may consider some sizable three-dimensional space of interest. This space may be any which is relevant to aero-applications such as over a field of crops or a military parachute drop zone. Aircraft vehicles, in this case, specifically designed to sample the atmosphere, can obtain velocity measurements over this three-dimensional space at any desired location. These data measurements may then serve as the boundary conditions demonstrated in Figure 1, or as a basis for some other numerical method. The key point is that data are not only available on the exterior of the domain but also in the interior.

Figure 1.

An illustration conveying the availability of data. In a solid steel rod (left), data are only available on the surface. For a windfield, data are available on the boundary as well as throughout the entire geometry. This allows for additional freedom in choice of boundary and interior conditions.

Windfields approximately based on potential flow (Ross et al., 1988), perhaps along with acoustic fields governed by the Helmholtz equation, represent rather unusual instances where sensor data from the interior of a domain can be used in conjunction with the free-space Green’s function to provide the given field solution throughout the domain. Rather than having boundary conditions specified, all boundary data could be determined from a system developed from the interior integral identities. This process will be referred to as the method of interior equations (MIEs).

The present effort is focused on the development of windfield estimation methods subjected to time constraints. For these cases, both computer modeling and data acquisition must proceed with high rapidity. These constraints suggest the use of a surrogate model based on potential flow for which a Green’s function formulation may be leveraged to reduce sampling requirements.

The potential flow assumption, acting as a surrogate model for wind, effectively replaces multi-disciplinary physics models, which will require a three-dimensional discretization. While this is likely to result in some loss of fidelity in windfields which are not purely harmonic, this allows the user to gain computational efficiency, and significantly reduces time and resources needed to collect physical data. The main attraction of the approach discussed in this report is that it allows a fast and reasonable approximation for a windfield, which is likely to be appropriate for many basic applications such as the aforementioned areas (strategic parachute drops and dispersal of chemical agents in both fire fighting and agriculture). Mission success may depend most strongly on the timeliness of supporting mission data to such an extent that the proposed method represents one of very few feasible approaches. It should be noted that potential flow assumptions are widely used in engineering, for example, in airfoil design (Anderson, 2010), noting that a high level of fidelity is required. This idea is extended for use as a surrogate model for wind, where fidelity requirements may be less stringent.

Numerical method and solution process

Numerical solution to Green’s third identity

For reference, Green’s third identity and its gradient are given as follows (Kellogg, 2012)

\begin{array}{l} ϕ (p) = \iint (ψ (p, Q_{k}) \frac{d ϕ (Q)}{d \hat{n}} - ϕ (Q) \frac{d ψ (p, Q_{k})}{d \hat{n}}) ds (Q) \\ ϕ {(p)}_{, i} = \iint (- ψ {(p, Q_{k})}_{, i} \frac{d ϕ (Q)}{d \hat{n}} + ϕ (Q) \frac{d ψ {(p, Q_{k})}_{, i}}{d \hat{n}}) ds (Q) \end{array}

(1)

where p represents a source-point location not coincident with the domain boundary, and Q represents a field-point location on the domain boundary. Capitalization is used to denote points that are on the boundary. In most cases, the boundary form of equation (1) forms the basis for numerical methods known collectively as the boundary element method (BEM). In discretizing equation (1), the following is typically performed:

The surface is discretized into a series of iso-paramentric elements over which an assumed interpolation is taken, both for the geometry of the surface and for variation of boundary data.

The interpolations are expressed in a common natural coordinate system, transforming the integrals.

The integrals are determined, either from closed-form integration formulae or approximated by Gaussian quadrature.

The discretized form of equation (1) may be given as

\begin{array}{l} ϕ (p) \approx \sum_{j = 1}^{NE} \sum_{k = 1}^{NGP} {[{(\sum_{n = 1}^{NPE} N_{n} (ξ_{k}, η_{k}) (ψ (p, Q_{k}) {\frac{d ϕ (Q)}{d \hat{n}} |}_{n} - ϕ {(Q)}_{n} \frac{d ψ (p, Q_{k})}{d \hat{n}}))}_{k} J_{k} W_{k}]}_{j} \\ ϕ {(p)}_{, i} \approx \sum_{j = 1}^{NE} \sum_{k = 1}^{NGP} {[{(\sum_{n = 1}^{NPE} N_{n} (ξ_{k}, η_{k}) (- (ψ {(p, Q_{k})}_{, i}) {\frac{d ϕ (Q)}{d \hat{n}} |}_{n} + ϕ {(Q)}_{n} \frac{d ψ {(p, Q_{k})}_{, i}}{d \hat{n}}))}_{k} J_{k} W_{k}]}_{j} \end{array}

(2)

In the above expression, $NE$ denotes the number of elements, $NGP$ denotes the number of Gauss points, $NPE$ denotes the number of nodes per element, and $J$ and $W$ denote the Jacobian of the transformation and the weight in the quadrature, respectively. The interpolation functions are denoted as $N_{n} (ξ, η)$ .

In equation (2), the unknowns are $\frac{d ϕ (Q)}{d \hat{n}}$ and $ϕ (Q)$ . One should note that these terms are denoted with a capitalized $Q$ , indicating that these values are located on the boundary. Each unknown term will have a contribution from the integrals associated with it, denoted as $A_{i j}$ . Equation (2) is then developed for multiple p-points producing a linear system of equations

[\begin{matrix} A_{11} & \dots \\ \dots & A_{i j} \end{matrix}] {\begin{matrix} \frac{d ϕ {(Q)}_{1}}{d \hat{n}} \\ ⋮ \\ \frac{d ϕ {(Q)}_{n}}{d \hat{n}} \\ ϕ {(Q)}_{1} \\ ⋮ \\ ϕ {(Q)}_{n} \end{matrix}} = {\begin{matrix} ϕ,_{i} (p_{1}) \\ ⋮ \\ ϕ,_{i} (p_{2 n}) \end{matrix}}

(3)

The Method of Interior Equations

The MIEs is a method used to solve for unknown boundary data and is the method that this report will introduce. The windfield problem is posed in such a way that velocity data are available both on the boundary of the domain and in the interior volume of the domain. This has a significant implication in terms of data collection.

If posed as a standard boundary integral problem, some combinations of $ϕ$ and $\frac{d ϕ}{d \hat{n}}$ are known on the boundary of the domain. This gives rise to multiple options, including the standard BEM (p-points selected on the boundary), or the method of fundamental solutions (MFS; Chen et al., 2008; Fairweather and Karageorghis, 1998; Kupradze, 1965) in which p-points are selected outside of the domain, where $ϕ (p)$ is identically zero due to the jump-discontinuity present in the Green’s function.

However, due to the nature of the windfield problem, data are available for collection on the interior volume of the domain (as opposed to solid-geometry problems). When considering the feasibility of real-world data collection, the time needed to obtain interior data drops drastically. Conditions are no longer needed on the boundary, nor needed at specific locations. A UAV may fly in a general area within the domain and collect velocity data at many locations. Thus, the importance of gathering data at specific nodes is no longer necessary. This is likely to reduce many problems in UAV path-planning for data collection.

Mathematically, this indicates that the boundary conditions are replaced by interior conditions. The applications of this type of condition are limited in scope and are unlikely to be applicable to any problems involving solid geometries. However, for potential problems involving fluid flow as well as acoustics, it is conceivable that interior conditions can be used. For visualization, Figure 2 is provided.

Figure 2.

A visualization of the MFS compared to the MIE. The MFS (left) requires p-points to be selected outside of the domain, where a part of the boundary solution is known, and the other part is obtained via solving a linear system. The MIE (right) requires the p-points to be selected inside the domain, where the value for the potential at each point is observed as raw data. The entire boundary solution is unknown when using the MIE.

Obtaining a solution

The boundary data are obtained from solution of the resulting linear system, giving the values of the potential function and its normal derivative at the nodes on the surface. Once the boundary solution is obtained, post-processing provides interior values of the potential function using equation (2) for any desired p-point. A diagram of the process for obtaining a solution is included in Figure 3.

Figure 3.

The process for reproducing a windfield is visualized. Interior data measurements (wind velocities) are first obtained from within the domain. These data are needed to obtain the boundary solution (velocities at nodal locations on the surface of the domain). The boundary solution is then used to post-process wind velocities at any desired location within the domain.

Figure 3 demonstrates the general steps for obtaining a windfield solution using the MIE:

Begin with known values of $\vec{\nabla} ϕ$ corresponding to the local wind-velocity components inside of the solution space and one condition on the arbitrary constant involved in $ϕ$ .

Mesh the exterior of the domain of interest. In the case of this report, the domain is a perfect cube (three-dimensional).

Write equation (3) using the p-points and the nodes of the exterior mesh. The right-hand side (RHS) of equation (3) is known from the interior data (step 1), and the $A_{i j}$ matrix is also known from geometry.

Solve equation (3) for the unknown vector. The vector is the boundary solution, which is the collection of $ϕ$ and $\frac{d ϕ}{d \hat{n}}$ at each boundary node. The location of these nodes are defined by the user’s mesh.

Equation (2) is then written for a new set of p-points. The windfield at any p-point within the domain may then be expressed using equation (2). At this point, the user may select any p-point of interest and obtain a solution for $ϕ$ or $\vec{\nabla} ϕ$ at that location.

Modeling details

Meshing and domain

In order to characterize a three-dimensional windfield, an appropriate geometry must be chosen. For problems involving flight within close proximity to the surface of the earth, a complex terrain is a likely candidate. However, for large open spaces over flat terrain such as in crop-dusting applications, a cube may be considered to be appropriate.

In test simulations involving transcendental functions which share the common feature of real wind, namely, that they cannot exactly be represented by a finite set of polynomials, triangular octic (eighth-order) elements will be used having 45 nodes per element. Figure 4 shows the nodal locations for a single element.

Figure 4.

A mesh of a single face of the domain model is provided. This provides a visualization for individual elements, which have a total of 45 nodes per element. For the global windfield domain, 12 octic elements are used.

In addition, the mesh for the entire cubic domain is provided for visualization. It should be noted that the mesh shown in Figure 5 is a unit cube of edge size 2, with a center located at (0, 0, 0). For the particular case of cube, a unit mesh is stored, so that the physical mesh coordinates may be multiplied by the size of the desired domain.

Figure 5.

The global mesh for the windfield domain is provided. Unit elements are used, with red points representing boundary nodes. There are 12 elements in total, each element containing 45 nodes.

For the present simulations, the mesh has been stretched by a factor of 500, resulting in a cube of edge size 1000. This will represent a 1000 × 1000 × 1000 m³ in wind simulations, resembling a space over a field of crops, or drop zone.

Location of p-points

When using the MIE for numerical simulations, the location of p-points is important to obtain an accurate solution. During development, it was observed that certain sets of p-points provide more numerical stability than others. Specifically, if p-points were placed too close, or too far from the domain boundary, the accuracy of the solution began to degrade. This is consistent with restrictions of p-point placement in the MFS, notably the so-called “fictitious boundary” (Golberg and Chen, 1998). The fictitious boundary is often employed to both avoid singularities arising by placing p-points on the boundary and to provide numerical stability with strategic p-point placement.

The present approach generates random locations for the interior p-points in an effort to replicate the scatter in data arising from physical measurements. Following extensive numerical investigation involving random sets of points, it was determined that accurate results were consistently obtained when points were taken from a region defined as the volume external to a cube 70% of the side length of the physical region and internal to a cube 85% of the side length of the physical region as shown in Figure 6. Locations of corner coordinates are given in Table 1. The locations shown in the table are prior to geometric stretching by the aforementioned factor of 500.

Figure 6.

An suitable region for p-points within the global mesh is shown. The outer most cube represents the domain boundary and the two inner cubes are used to demonstrate an bounded volumetric region for the placement of p-points.

Table 1.

Corner locations for each cube.

No.	Cube 1	Cube 2	Cube 3
Corner 1	(−1, 1, 1)	(−0.85, 0.85, 0.85)	(−0.7, 0.7, 0.7)
Corner 2	(1, 1, 1)	(0.85, 0.85, 0.85)	(0.7, 0.7, 0.7)
Corner 3	(−1, 1, −1)	(−0.85, 0.85, −0.85)	(−0.7, 0.7, −0.7)
Corner 4	(1, 1, −1)	(0.85, 0.85, −0.85)	(0.7, 0.7, −0.7)
Corner 5	(−1, −1, 1)	(−0.85, −0.85, 0.85)	(−0.7, −0.7, 0.7)
Corner 6	(1, −1, 1)	(0.85, −0.85, 0.85)	(0.7, −0.7, 0.7)
Corner 7	(−1, −1, −1)	(−0.85, −0.85, −0.85)	(−0.7, −0.7, −0.7)
Corner 8	(1, −1, −1)	(0.85, −0.85, −0.85)	(0.7, −0.7, −0.7)

A suitable set of locations for p-point placement is the volume between cube 2 and cube 3; the global domain boundary is cube 1. This table corresponds with the geometries in Figure 6.

If the boundary of the domain (cube 1) is referred to as a “cube of size 1,” then the region of acceptable p-point placement is bounded by a cube of size 0.85, and a cube of size 0.7. The p-point mesh used for MIE simulations is a randomly generated set of points placed on the surface of a cube of size 0.8 which lies within the suitable volume for p-points. All of these meshes were stretched by a factor of 500 and all meshes are centered at (0, 0, 0).

Description of computer simulation

To perform the numerical experiment, a Fortran computer program was written and compiled with the gfortran compiler. This program requires a mesh of the boundary domain, a p-point mesh, values for the potential function at the p-points, and a post-processing mesh. After building and solving the linear system, the program uses the computed boundary to post-process velocity values at any desired point within the domain. All simulations were run on an ACER Aspire V 15 (Intel Core i7-5500U CPU at 2.40 GHz, 8 GB DDR3 L Memory, LINUX environment). The program has several options including various quadrature schemes, a library of linear through duodecic elements, and the option to accept wind data in excess of the linear system rank consistent with anticipated UAV data. Simulation run times depend on the choice of quadrature, element order, and the number of interior data.

For the test cases to be presented in subsequent sections, two different program settings were chosen. The use of sextic elements and one linear system was observed to accurately characterize any harmonic function which is well approximated as a sixth-order polynomial, with run times in the order of 50 s which is suitable for most real-time applications. These settings were used for the case of sixth-order polynomial potential shown in equations (4) and (5). Simulations for the transcendental potential also shown in equations (6) and (7) were based on octic elements and no redundant p-points.

Results

The present effort has been in anticipation of characterizing a physical windfield, by performing an experiment to sample the atmosphere. Future work is planned to acquire the necessary hardware, facilities, and funding to obtain the necessary physical measurements needed. At this time, results for a theoretical test case are provided.

The eventual application of the current method will require a fleet of UAVs equipped with suitable instrumentation and data acquisition hardware. A similar data collection method has been presented in Montalvo et al. (2015), which was designed for UAVs to sample mean wind speed. Using approximately 20 commonly available UAVs, it is estimated to take approximately 10 min to obtain a volumetric sample of a windfield in the order of 1 km³. The technique proposed in this report will require significantly fewer UAVs and less time, as model requirements only require surface unknowns through the use of the Green’s function. In addition, large spatial changes in wind environments are in the order of hours (Etkin, 2012), which is consistent with this technique. The current method is benchmarked using arbitrary potential fields with significant spatial variation in terms of high-order polynomials and transcendental functions.

The results provided for the MIE are for a case in which data are sampled from a known potential function, and then post-processed at p-points which were not used in the linear system. In practice, the generating function will not be known, but as long as post-processing shows strong results, the function is well characterized by the numerical solution of the integral identity (equation (1)).

The conditions for the case of the MIE are consistent with the conditions of physical data collection. That is, $\vec{\nabla} ϕ$ is known at observed locations (p-points) on the domain interior. These points are referred to as interior conditions. The potential function, $ϕ$ , and its normal derivative, $\frac{d ϕ}{d \hat{n}}$ , are unknown at all locations on the boundary. The interior conditions effectively replace boundary conditions.

When performing numerical experiments for different types of potential functions, it was observed that the boundary solutions for potentials which can be exactly modeled by polynomial shape functions show different behavior from those for which the shape functions cannot fully model their behavior. However, regardless of the form of the potential, post-processing interior results are extremely accurate when using the MIE. As an example, results are provided for two cases of investigation: a sixth-order polynomial potential and a nonlinear, transcendental potential.

Sixth-order polynomial potential function

Numerical examples are chosen based on two functions recognized as exact solutions of Laplace’s equation. The two functions chosen are thought to represent a more formidable modeling challenge than what would be encountered in a potential solution describing a windfield. The first is a sixth-order harmonic polynomial function

\begin{matrix} ϕ = - κ (15 z^{6} - (105 y^{2} + 120 x^{2}) z^{4} + (105 y^{4} + 120 x^{4}) z^{2} - y^{6}) \\ - κ (90 x^{2} y^{4} + 90 x^{4} y^{2} - 14 x^{6}) + x^{2} + y^{2} - 2 z^{2} \end{matrix}

(4)

and its gradient is

\begin{matrix} \vec{\nabla} ϕ = (2 x (1 + 6 κ (7 x^{4} - 10 x^{2} (3 y^{2} + 4 z^{2}) + 5 (3 y^{4} + 4 z^{4})))) \hat{i} \\ + (2 y (1 + 3 κ (- 30 x^{4} + 60 x^{2} y^{2} + y^{4} - 70 y^{2} z^{2} + 35 z^{4}))) \hat{j} \\ + (- 2 z (2 + 15 κ (8 x^{4} - 16 x^{2} z^{2} + 7 y^{4} - 14 y^{2} z^{2} + 3 z^{4}))) \hat{k} \end{matrix}

(5)

where $κ {= 10}^{- 12}$ is a geometric scaling factor. This function was sampled to generate the results shown in Figures 7 and 8.

Figure 7.

A comparison of the true and numerical boundary solutions for equation (4), using the MIE to compute numerical results. Here, z = 500, corresponding to the top surface of the domain. The continuous surface represents the true potential function, while the discrete points (yellow) represent numerical, nodal values computed with the MIE. It is clear that $ϕ$ is well captured here.

Figure 8.

A comparison of the true and numerical boundary solutions for equation (5), using the MIE to compute numerical results. Here, z = 500, corresponding to the top surface of the domain. The continuous surface represents the true potential function, while the discrete points (yellow) represent numerical, nodal values computed with the MIE. It is clear that $\frac{d ϕ}{d \hat{n}}$ is well captured here.

There should be no advantage expected in using elements higher than sextic to model a sixth-order polynomial potential. Hence, sixth-order shape functions were used to model this potential, rather than the octic elements used in the next example. Figures 7 and 8 demonstrate that the computed boundary solution obtained for $ϕ$ and $\frac{d ϕ}{d \hat{n}}$ , respectively, captures a sixth-order polynomial potential function extremely well. This demonstrates that for a high-order polynomial potential function which is completely captured by the element shape functions, an accurate boundary solution is obtained using the MIE.

In addition to visual results, errors of interest have been included in Table 2. This report defines normalized error as

Normalized error = {(\frac{ϕ_{true} - ϕ_{numerical}}{ϕ_{largest}})}_{i}

where $ϕ_{largest}$ is the largest true value from the data set.

Table 2.

Errors of interest seen in boundary solution results associated with the collective boundary solution visualized in Figures 7 and 8.

Parameter	Value
Correlation coefficient ( $R$ )	0.999999999999978
Highest normalized error observed	2.982 ${(10}^{- 7})$
Lowest normalized error observed	1.857 ${(10}^{- 12})$
Average normalized error observed	3.657 ${(10}^{- 8})$
Maximum absolute difference	0.2782
Minimum absolute difference	6.59 ${(10}^{- 6})$
Average absolute difference	0.3412 ${(10}^{- 1})$

These errors represent a comparison between the true and numerical solutions of equations (4) and (5). Numerical solutions were computed using the MIE.

Table 2 shows that for a field variable of order 10⁶, the maximum difference observed is of the order 10⁻¹, while obtaining an average accuracy of 10⁻². This indicates that on average, a boundary solution computed with the MIE is expected to show approximately eight significant figures of accuracy, for the case of a polynomial potential.

Transcendental function

A potential involving products of transcendental functions was used to provide sampled data for the MIE. The rationale for choosing this potential is to ensure that both the computer simulation and numerical method can accurately characterize a function that is likely to be more formidable than the potential functions observed in most problems arising in physics.

The harmonic function used for verification is given as

ϕ = e^{0.003 z} \sin (14 + 0.003 x) + e^{0.003 x} \sin (14 + 0.003 y) + e^{0.003 y} \sin (14 + 0.003 z)

(6)

and its gradient is

\begin{matrix} \vec{\nabla} ϕ = (0.003 e^{0.003 z} \cos (14 + 0.003 x) + 0.003 e^{0.003 x} \sin (14 + 0.003 y)) \hat{i} \\ + (0.003 e^{0.003 x} \cos (14 + 0.003 y) + 0.003 e^{0.003 y} \sin (14 + 0.003 z)) \hat{j} \\ + (0.003 e^{0.003 y} \cos (14 + 0.003 z) + 0.003 e^{0.003 z} \sin (14 + 0.003 x)) \hat{k} \end{matrix}

(7)

It is noted that the second example involving transcendental functions cannot be exactly modeled by any finite set of polynomial shape functions. Here, it is modeled using eighth-order shape functions (octic elements, as previously discussed in the “Modeling details” section). The formulation presented here never directly addresses the unknown boundary data and so, most generally, it is similar to the indirect BEM (Banerjee and Butterfield, 1981) which is also known by various other names such as the fictitious BEM and the unknown surface density method. While the previous example showed that the boundary solution took the form of the interior solution when it could be exactly modeled by the shape functions, the present example includes some portion of the non-physical solution. With regard to this, it is mentioned here that the present formulation could possibly be of interest in various comparison studies between the direct and indirect BEM.

Since the boundary solution for the transcendental case has at least some component of the non-physical boundary solution, characterization of its success in modeling a three-dimensional windfield is given based on interior post-processing. Post-processing was performed on a cubic grid of 64 nodes and the results are shown in Figure 9.

Figure 9.

Post-processing vector results for wind-velocity computed at interior locations.

In addition to Figure 9, numerical errors of interest are included in Table 3 to demonstrate post-processing accuracy.

Table 3.

Errors of interest seen in post-processing results (MIE).

Parameter	Value
Correlation coefficient ( $R$ )	0.999999995
Highest normalized error observed	5.366 ${(10}^{- 5})$
Lowest normalized error observed	2.47 ${(10}^{- 15})$
Average normalized error observed	2.484 ${(10}^{- 6})$
Maximum absolute difference	5.799 ${(10}^{- 7})$
Minimum absolute difference	2.7 ${(10}^{- 16})$
Average absolute difference	2.813 ${(10}^{- 8})$

These errors represent the numerical solution on the interior of the domain.

Table 3 shows that for a field variable of order 10⁻², the maximum difference observed is of the order 10⁻⁷, while obtaining an average accuracy of 10⁻⁸. This means that on average, a solution will show approximately six significant figures of accuracy in post-processing with the MIE and, in some cases, will show 14 significant figures of accuracy.

Conclusion

For applications where the interior of a domain is of primary interest, and where limited data from measurements are available, the MIE offers a fast method to reproduce a given field throughout the domain. Notable examples of such problems include those in wind-mapping and in acoustics. Scaling of the sampling data required is proportional to surface densities and not volume densities, reducing overall problem dimensionality by 1 which is a well-known characteristic of BEMs. A test case showed that the MIE can produce 14 significant figures of accuracy through post-processing while providing sufficient speed on a laptop computer to provide an effective means to support near real-time model-based decision-making using a limited set of data. The work presented here is an initial step toward development of a system to characterize actual windfields based on limited physical measurements.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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