Force and moment partitioning influence potential solver 2D

Computes 2-D influence potential fields of the force and moment partitioning method using the MATLAB Partial Differential Equation Toolbox.

fmpmPotentialPDE2d computes the influence potential field of the

force and moment partitioning method [1] using the MATLAB Partial

Differential Equation Toolbox.

Two-Dimensional version developed based on code used in Zhu et al. [2].

The boundary conditions on the non-reference geometry are

(n dot grad(phi)) = 0 in multi-body configurations, which implies the

non-penetration property of the influence field.

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Syntax:

phi = fmpmPotentialPDE2d(xdomain, ydomain, geo, 'loadtype')

phi = fmpmPotentialPDE2d(xdomain, ydomain, geo1, geo2, ..., geoN, 'loadtype')

phi = fmpmPotentialPDE2d(__, refgeoN)

phi = fmpmPotentialPDE2d(__, refgeoN, innerTrSize, outerTrSize)

[phi, X, Y] = fmpmPotentialPDE2d(__)

[__, bodygeo] = fmpmPotentialPDE2d(__)

[__, bodygeo, x_surf, y_surf] = fmpmPotentialPDE2d(__)

[__, bodygeo, x_surf, y_surf, phi_surf] = fmpmPotentialPDE2d(__)

[__, bodygeo, x_surf, y_surf, phi_surf, dphidx_surf, dphidy_surf] = fmpmPotentialPDE2d(__)

[__, bodygeo, x_surf, y_surf, phi_surf, dphidx_surf, dphidy_surf, norm_surf] = fmpmPotentialPDE2d(__)

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Inputs:

xdomain: x-coordiante vector of the computational domain (outter boundary).

ydomain: y-coordiante vector of the computational domain (outter boundary).

geo1, geo2, ..., geoN: N-by-2 vectors give the x and y coordinates of the

body geometries. (inner boundary). The first

geometry geo1 is the reference geometry for which

the phi field will be computed by default. The

referece geometry/geometries can be specified by

the refgeoN argument otherwise.

'loadtype': a char variable that specifies the type of load of which the

influence field will be computed. 'loadtype' must be 'lift',

'pitch', or 'drag'. When 'pitch' is used, a 1-by-2 array,

refaxis, that specifies the x and y coordinates of the pitch

moment reference axis should follow the 'pitch' argument. If

refaxis is empty [], the default refaxis is the geometric

center of the reference geometry.

refgeoN: a scalar or a row vector that specifies the reference

geometry/geometries for the phi field. The values of the

elements refer to the order of geo1, geo2, ..., geoN inputs. The

default reference geometry is geo1 if refgeoN is not specified

or refgeoN = [];

innerTrSize: a scalar that specifies the size of the triangular mesh on

the inner boudaries (bodies). The default value is 0.5*ds

where ds is the minimum unit length of the computational

domain if innerTrSize is not specified or innerTrSize = [];

outerTrSize: a scalar that specifies the size of the triangular mesh on

the outer boudaries (computational domain). The default

value is 10*ds where ds is the minimum unit length of the

computational domain if outerTrSize is not specified or

outerTrSize = [];

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Outputs:

phi: influence potential field.

X: x-coordinate mesh generated based on xdomain and ydomain.

Y: y-coordinate mesh generated based on xdomain and ydomain.

bodygeo: polyshape object defined by the geo input.

x_surf: x-coordinate vector of the data points on the reference body

surface.

y_surf: y-coordinate vector of the data points on the reference body

surface.

phi_surf: phi vector of the data points on the reference body surface.

dphidx_surf: vector of gradients of phi in the x-direction on the data

points on the reference body surface.

dphidy_surf: vector of gradients of phi in the y-direction on the data

points on the reference body surface.

normxy_surf: an N-by-2 array that contains the x- (first column) and y-

(second column) componets of the surface normal vectors of

the reference geometries. The dimension and the order of

normxy_surf corespond to x_surf and y_surf.

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Examples:

1. geo = [[-0.5, 0.5, 0]', [-0.5, -0.5, 0.5]'];

[phi, X, Y] = fmpmPotentialPDE2d(-2:0.1:2, -1:0.1:1, geo, 'lift');

Computes the lift influence potential field on a triangle defiend by

points (-0.5, -0.5), (0.5, -0.5), and (0, 0.5) in a computational

domain defiend by x-coordinates -2:0.1:2 and y-coordinates -1:0.1:1.

Output phi returns the influence field; X and Y return the mesh grids.

2. geo1 = [[-0.5, 0.5, 0]', [-0.5, -0.5, 0.5]'];

geo2 = [[0, 1.2, 1.2, 0]', [-0.2, -0.2, 0.1, 0.1]'];

[phi, ~, ~, bodyshape] = ...

fmpmPotentialPDE2d(-2:0.1:2, -1:0.1:1, geo1, geo2, 'pitch', [0.5, 0], 2, 0.05, 0.1);

Computes the pitch moment influence potential field on geo2

(as refgeoN = 2) in a multi-body geometry by geo1 and geo2. The size

of the triangular mesh on the geometry is 0.05 on the inner boundaries

and 0.1 on the outer boudaries. The pitch moment reference point is

given by refaxis = [0.5, 0]. Output bodyshape returns the combined

geometry as a polyshape object.

3. geo1 = [[-0.5, 0.5, 0]', [-0.5, -0.5, 0.5]'];

geo2 = [[0, 1.2, 1.2, 0]', [-0.2, -0.2, 0.1, 0.1]'];

geo3 = [[0.1, 1.3, 1.3, 0.1]', [-0.3, -0.3, 0, 0]'];

phi = fmpmPotentialPDE2d(-2:0.1:2, -1:0.1:1, geo1, geo2, geo3, 'pitch', [], [1, 3]);

Computes the pitch moment influence potential field on the combined

body of geo1 and geo3 (as refgeoN = [1, 3]) referenced to the

geometric center (as refaxis = []) of the combined geometry.

4. geo1 = [[-0.5, 0.5, 0]', [-0.5, -0.5, 0.5]'];

geo2 = [[0, 1.2, 1.2, 0]', [-0.2, -0.2, 0.1, 0.1]'];

[phi, ~, ~, bodyshape, x_surf, y_surf, phi_surf, ~, ~, normxy_surf] = ...

fmpmPotentialPDE2d(-2:0.1:2, -1:0.1:1, geo1, geo2, 'lift');

Computes the lift influence potential field of geo1 (default setting

as refgeoN is not specified) and returns the polygon object of the

conbined geometry that contains geo1 and geo2. x_surf and y_surf

returns the coordinates of the surface data points on the reference

geometry geo1. phi_surf returns the corresponding phi values and

normxy_surf returns the corresponding surface normal vectors of these

points.

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version 2.0.1

- Fixed a bug where using a single output argument causes a formatting

error.

Xiaowei He and Yuanhang Zhu

7/20/2023

% -------------------------------------------------------------------------

version 2.0.0

- Extended the influence field to multi-body configurations.

- Added an output argument for surface normal vectors on the reference

geometry.

- Optimized the organization of the input and output arguments

accordingly.

- Updates in the headlines and examples.

Xiaowei He and Yuanhang Zhu

7/17/2023

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version 1.2.1

- Optimized the outer boundary detection to prevent potential false

boundary conditions.

- Changed the default minimum triangular size to 0.5*innerTrSize to

prevent potential insufficient element size.

Xiaowei He and Yuanhang Zhu

6/19/2023

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version 1.1.0

- Added output options including phi values on the body surface and the

corresponding x- and y- gradients. See e.g.3 and e.g.4.

Xiaowei He and Yuanhang Zhu

6/15/2023

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version 1.0.0

Xiaowei He and Yuanhang Zhu

05/30/2023

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[1]. Menon, K. & Mittal, R. (2021). Quantitative analysis of the

kinematics and induced aerodynamic loading of individual vortices in

vortex-dominated flows: A computation and data-driven approach. Journal

of Computational Physics 443, 110515.

[2]. Zhu et al. (2023). Force moment partitioning and scaling analysis of

vortices shed by a 2D pitching wing in quiescent fluid. arXiv:2301.13373

[physics.flu-dyn].

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