A New Advancement in the Complex Variable Boundary Element Method with a Potential Flow Application

Transcription

1 A A New Advancement in Complex Variable Boundary Element Method with a Potential Flow Massachusetts Institute of Technology, United States Military Academy bdwilkins95@gmail.com November 2, 2018

2 A Overview

3 A Section 1

4 A Potential Flow Around a Cylinder We consider flow of an ideal, irrotational, incompressible fluid around a cylinder. Characteristics of this particular flow: Far away from cylinder, flow is unidirectional and uniform. The flow has no vorticity, and so velocity field V is irrotational, hence V = 0. Being irrotational, re exists a velocity potential ϕ satisfying V = ϕ. Being incompressible, V = 0. We conclude that ϕ must satisfy Laplace s equation, 2 ϕ = 0.

5 A Depiction of alytic Solution Figure: Zoomed-in Figure: Zoomed-out Two perspectives of analytic solution. Notice how flow approaches uniform flow at points farr away from location of cylinder.

6 A Due to symmetry, we can choose to model only upper half plane. We chose upper half plane as opposed to modeling just a single quadrant because we wanted approximation to include two stagnation points at (1, 0) and ( 1, 0). Boundary conditions came from imaginary part of complex variable function z + 1 z Modeling Considerations Figure: Vertical and horizontal axes of symmetry. The problem symmetry allows for modeling simplification.

7 A Section 2

8 A Fundamentals Theorem (The Cauchy Integral Theorem) Let Γ be a simple closed contour, and let f be analytic everywhere within and on Γ. Then, for any point, z 0 in domain enclosed by Γ, f (z 0 ) = 1 f (z)dz. (1) 2πi Γ z z 0 Figure: Boundary Value Problem Depicting Simple Closed Contour

9 A The General Approximation Function A approximation function is a linear combination of analytic complex variable functions. where ˆω(z) = n c j g j (z), (2) j=1 c j C are complex coefficients, g j (z) are complex variable basis functions being used in approximation, n is number of basis functions being used in approximation We note re are 2n degrees of freedom since each complex coefficient has an unknown real part as well as an unknown imaginary part.

10 A The Modeling Procedure 1 The begins by interpolating between given boundary values to generate a simple closed contour along boundary of problem domain. When linear interpolation is used, numerical integration of Cauchy integral equation yields basis functions of form: (z z j ) ln(z z j ) 2 The modeler selects z j that are to be used in approximation. These points are branch points of logarithm and are often referred to as nodes. 3 Collocation with known boundary conditions is used to determine coefficients of approximation function. 4 Once coefficients are known, approximate equipotential and stream lines can be evaluated continuously in plane as real and imaginary parts of function, respectively, at all points at which basis functions are analytic.

11 A In un-optimized implementation of algorithm, re is not much thought given to location of nodes and collocation points. We just chose locations that seemed reasonable. So, collocation points were equally spaced along problem domain, and nodes were arranged in a circle around problem domain. Example Problem Situation y Problem Geometry Problem Domain Collocation Points Nodes x Figure: Red dots correspond to collocation points, black dots are nodes.

12 A Section 3

13 A Description of Step 1 Many candidate nodes and collocation points are generated. Step 2 The algorithm is initialized by arbitrary selection of two collocation points. Step 3 Then, each of candidate nodes is checked to see which node results in 1-node model of least error given initial selection of two collocation points.

14 A Description of Step 4 The node corresponding to model of least maximum error is selected as next node. Step 5 With newly-selected node now incorporated in model, error is now re-assessed along entire boundary. The two locations of greatest error are selected as locations of next two collocation points. Step 6 Then, each of remaining candidate nodes is checked to see which node results in model of least error given previous selections of collocation points and nodes. Step 7 Steps 4-6 repeat until all required number of nodes and collocation points have been selected.

15 A Refinement Procedure At this point, we have selected all of nodes and collocation points that we need. But we can still reduce error in our approximation. Step 8 Starting with first selected node, each node is re-examined one-by-one to see if a different selection from pool of remaining candidate nodes would result in a model with smaller maximum error. Step 9 If selecting a different node would result in a smaller overall error, that node replaces currently-used node and we proceed to check next node. Orwise, currently-used node is kept and next node is checked. At this point, note that maximum error is decreasing monotonically since eir no change is made to model or a change resulting in a smaller maximum error is made. Step 10 Refinement continues for a pre-specified number of iterations or until maximum error is no longer decreasing.

16 A Section 4

17 A Parameters Example Problem Details Values Degrees of freedom: 40 Length: 20 Height: 16 Number of candidate nodes for optimization algorithm: 2000 Number of candidate collocation points for optimization algorithm: 1500 Table: Parameter Details Method Un-optimized: Optimized: Maximum Absolute Error Table: Error Details

18 y y A Flow net approximation in Optimized vs. Un-optimized y Flow net approximation in x x 1.5 Flow net in vicinity of obstacle 1.5 Flow net in vicinity of obstacle 1 1 y x x Figure: Optimized Figure: Un-optimized

19 A Flow net near left stagnation point at (-1,0) Optimized - Left Stagnation Point 0.8 y x Approximation alytic

20 A Flow net near right stagnation point at (1,0) Optimized - Right Stagnation Point 0.8 y x Approximation alytic

21 A y Optimized - Flow Near Boundaries Flow net near left boundary. y Flow net near right boundary x x At boundaries, flow approaches uniform flow, as desired.

22 A Optimized - Flow Near Boundaries y Flow net near top boundary x At boundaries, flow approaches uniform flow, as desired.