Proving The Solubility Of A Navier-Stokes Solution Using A Modified Method Of Particular Solutions

8/2/2019 Proving The Solubility Of A Navier-Stokes Solution Using A Modified Method Of Particular Solutions

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Department of Mechanical, Materials

& Manufacturing Engineering

3rd

Year Individual Project

MM3BPR

Tim Wong

4100673

Professor Henry Power

Proving The Solubility Of A Navier-Stokes Solution UsingA Modified Method Of Particular Solutions

BEng Mechanical Engineering2011/12


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Abstract

This report contains details on the use of the Modified Method of Particular Solutions

(MAPS) scheme, with the ambition of developing a more accurate and time efficient method

for solving two dimensional problems with regards to the Navier-Stokes equation forincompressible fluid flow. The scenario examined is a lid-driven cavity flow problem, with

the assumption of a linear superposition of the Stokes particular solutions, and an assumption

of non-linearity for the convective terms of the full Navier-Stokes equation, considered as a

non-homogeneous Stokes system, with the equation's base term ad Multiquadratic (MQ)

Radial Basis Functions (RBF) as source terms. These equations are then solved by Picard

iteration, and the resultant general solutions are independent of boundary conditions. Two

solvers are compared in the report, firstly a simple Gauss direct solver is, which is compared

with a Single Value Decomposition (SVD) method. The SVD solver was found to be more

accurate than the original solver, but less efficient on computing power.

The scheme can be considered mass conservative, as although the continuity equation is notexpressly imposed in the formulation of the scheme, the Particular solutions satisfy the mass

conservation equation already. In addition, pressure-velocity coupling is not an issue, as the

pressure particular solutions are found from the velocity particular solutions. The scheme

outlined above is then compared with pre-obtained results, calculated using the numerical

solutions of the analytical Kovasnay flow problem at different Reynolds Numbers, Re. (1)

Convergence and its dependence on the Shape Parameter is analyzed by finding the Root

Mean Square (RMS) error for the uniform distributions of collocation points; i.e. for several

pairs of nodal points, and the MQ Shape Parameter. The correlations show that the MAPS

scheme for both solvers is, for a wide range of Shape Parameters, generally stable when

compared with the pre-obtained results up to a Reynolds number of 3200. solutions can be

found for a range of nodal distributions, and increasingly accurate results can be found using

an SVD solver, at the expense of computing efficiency.


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Contents

Title Page .............................................................................................................................................. 1

Abstract................................................................................................................................................. 2

Nomenclature........................................................................................................................................ 4

Aims & Objectives ............................................................................................................................... 4

Introduction & Background .................................................................................................................. 4

The Modified Method Of Particular Solutions For Solving Navier-Stokes System Of Equations ........ 7

The Simple Gaussian Solver ......................................................................................................... 10

The Singular Value Decomposition Solver .................................................................................. 10

Pre-Estimated Solutions ............................................................................................................... 11

Lid Driven Cavity Flow Scenario Numerical Results ................................................................. 12

The Extension To, & Affects of Higher Reynolds Numbers ...................................................... 14

Computing Efficiency ................................................................................................................... 15

Conclusions ........................................................................................................................................ 16

Recommendations ............................................................................................................................... 16

Acknowledgements& References ....................................................................................................... 16

Bibliography ....................................................................................................................................... 17

Appendices ......................................................................................................................................... 18

Figures and Tables ...................................................................................................................... 18

MATLAB Code ............................................................................................................................. 18

Main Code .................................................................................................................................. 18

Memory Analysis Code .............................................................................................................. 20

Supervision Records ................................................................................................................... 20


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Nomenclature1

SP= Shape Parameter

MQ= Multiquadratic

RBF=Radial Basis Function

SVD =Singular Value Decomposition

Re = Reynolds Number

Aims & Objectives

The main aim of the project is to assess the possible improvements to be implemented to the

MAPS, relating to the solving of a two dimensional Navier Stokes Flow Problem.

Specifically;

Coding, Implementing and Analyzing the use of different Solvers within the Picard

iterations scheme Assess the sensitivity of the numerical solutions with respect to the value of the Shape

parameter

Comparing the Solvers for efficiency and range up to higher Reynolds Numbers

Producing results in good agreement with existing numerical data

Introduction & Background

Within the field of Computational Dynamics, it can be said that the numerical solution of

Navier-Stokes systems of equations are the most challenging to solve. This is when assuming

an incompressible fluid, and in terms of the basic variables p, . Schemes such as theSIMPLE (2) and PISO (3) types can transform the p- coupled non-linear problem into asystem of linear problems, these techniques have been used with wide success when using

standard methods such as Finite Volume (FVM) and Finite Element (FEM). However these

pairings require a large amount of computing effort in terms of power and time to solve the

systems of equations.

One solution to this is to avoid the p- coupling in 2-D Navier-Stokes systems, by usingvorticity formulations to decouple the pressure field from the governing equations. This line

of research has still not been fully explored, and still suffers from problems. A large

drawback is that the continuity equation is not explicitly imposed in the formulation, and

hence mass-conservation cannot be guaranteed. Some schemes, such as the Velocity

Correction Method (4) purport to correct the mass imbalance; however it seems that if

improved coupling strategies for the primitive variables can be implemented, then the

numerical solutions in terms of p- will provide more accurate pressure fields.Opposing this increase in accuracy however, for such methods as FEM, FD etc. as mentioned

above, is their reliance upon a computational mesh (5), which takes up considerable time and

effort to generate a high quality, accurate Mesh. To decrease the amount of effort and time

associated with the mesh generation, meshless numerical schemes have been developed,

allowing non-connected nodal distributions to be used, instead of connected elements. These

schemes such as the global Radial Basis Function (RBF) (6), used in the final scheme for thisproject, which uses a direct meshless collocation method, has had considerable success in

solving boundary value problems governed by Partial Differential Equations (PDEs). RBF

1 All other terms will be explained within the working present below


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collocation methods have an advantage in that they are completely meshless, and hence

dimensionally independent of the formulation, they also avoid the spectral convergence of

some of the RBF interpolants, e.g. MQ. Despite RBF global collocation methods inherent

advantages, they still suffer a fundamental problem in the form of the uncertainty relationship

(7). In essence this describes a relationship in which better conditioning is associated with

worse accuracy and vice-versa. System size is also a factor, as it is increased, the problem

becomes more troublesome. To combat this, approaches such as RBF-specific matrix

preconditioners (8) have been developed.

Despite the improvements in Global approaches, there are still many drawbacks to the

approach, matrix size is still a limiting factor, and ill conditioning still proves to a large issue.

Hence recent developments have suggested local RBF interpolation schemes to be more

efficient, and work for a greater range of shape parameter values. They also avoid many of

the ill conditioning problems at a local level. However, when using the local formulations,

the numerical scheme results in a looser meshless character as partial local connectivity is

need between the interpolation nodes, this results in additional, more complex programming.

Using an indirect approach, where RBFs can be used to approximate the complex high order

derivatives present in the PDE to be solved, provides more stability and reduces convergence

issues. This type of scheme integrates the approximated derivatives with respect to the

corresponding coordinates, (9) & (10) and only the collocation points along the line of

integration are considered to express a local formulation. This allows accurate solutions to be

obtained for a greater range of shape parameter values, as the integration process allows for

smoother behavior.

RBF meshless numerical solutions of Navier Stokes systems of equations use velocity or

stream function and vorticity formulations, to avoid the problem of the p- couplingformulation. Despite some success (11) in using a global approach, due to the aforementionedill conditioning problems, a local approach has been much more successful at predicting

viscous flow fields. The MQ Differential Quadrature method (MQDQ)(12) (13)has been

proposed for spatial discretization, and combined with the fractional step method, can be used

to couple the pressure and velocity in transient problems, obtaining good results. It was been

found however, that it is important to find an efficient way of finding the correct value of the

shape parameter (SP), as this is a highly critical value relating to the accuracy of the final

numerical solution. Work by Divo and Kassab(14) and Sanjasiraju and Chandhini(15) has

developed explicit locally based transient time stepping algorithms(16) for the p-formulation (14), overall combined with either the fractional step method, mentioned above,

or the Mark and Cell (MAC) method, and provided good results. The locally explicit

transient schemes allowed the results to be more stable and less affected by the SP. Workmentioned above into Global schemes (11) has been extended into a more local approach,

using a weighted time stepping algorithm, The Crank-Nicholson, which uses a simple Picard

iterative scheme. This proved more than enough to deal with the produces system of non-

linear equations. When compared with the Global scheme, results were similar, but required

far less computational time and effort. Added to these developments, a RBF interpolation

scheme (4), again working with a velocity-vorticity relationship, has used a velocity

correction method to solve many of the mass conservation problems inherent in this scheme.

Excellent results have been found up to Reynolds numbers of 10,000 using a nodal

configuration of 161x161.

Another recent development has been of an integrated RBF method (17) for use with linear

PDEs whose differential operator L(.()), can be wholly or partially expressed as a radialcomponent in polar or spherical coordinates;


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Equation 1

= 1+ 2 =()

It was suggested to approximate the radial component of the PDE (1. ), in terms of anRBF interpolation, and to then integrate the produced non-homogenous ODE. So producing

an approximate field variable representation by superposition of the corresponding particular

solution, similar to the indirect scheme proposed (9). The difference being that rather than

only the higher order derivatives being approximated by integration by a Cartesian grid, the

complete integration of the L1r operator with no reference to any grid is found, resulting in;

Equation 2

1 = ()

=1

With corresponding field variables for use

with linear PDEs whose differential

operator L(.(), can be wholly or partiallyexpressed as a radial component in polar or

spherical coordinates;Equation 3

= (

=1).

In this way the particular solution

(

) is given by the solution of the non-homogeneous

equation (below, equation 4), using the RBF () as the non-homogeneous term;Equation 4

1 = ()By using substituting the () approximation into the full () and the boundaryconditions, taking into account equation 2, one can obtain a fully meshless integrated RBF

solution of the boundary value problem, with no relations to any form of Cartesian Grid. This

approach is the Method Of Approximated Particular Solutions (MAPS).

For the case of a linear boundary problem;

Equation 5

( =, where &

Equation 6

( = , where Again, as above, the PDEs L(u) operator is expressed only in terms of radial coordinates,

plus a boundary operator B. These are then reduced to find the solution of the produced linear

system of equations;

Figure 1: Showing the Proposed Solution Domain, and the

Internal () and Boundary () Nodes


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Equation 7

This is valid for N points, (Nb + Ni = N, Boundary points and Internal points respectively), as

shown on Figure 1. The solution is found by solving for the values of. The solution

processes (or solvers) are discussed in a later section.

For this paper, the method proposed is a global approach, using a meshless collocation

method based on the MAPS. Intending to solve a two-dimensional system of Navier Stokesequations, for a p- formulation, this should allow the ability to predict flow fields for a goodrange of medium and high Reynolds numbers, whilst avoiding matrix ill conditioning

problems. Tor the approximations of the non-linear inertial terms, a simple Picard iteration

scheme is to be used, it is favored over more complex non-linear solvers as it is found to

provide an acceptable level of accuracy without having to further complicate the computing

efforts. To predict the approximate velocity and pressure fields, a Stoke Particular Solution is

found using Oseens decomposition formula (18). This fills the need to find a close form

expression for the particular solution.

A p-

coupling strategy is not needed in the proposed scheme, as the velocity particular

solution satisfies the continuity equation exactly, and as the particular solution for thepressure is found directly from the velocity solution, it is also satisfactory.

The Modified MAPS scheme proposed is compared for both solvers, against pre-existing

numerical results for the Kovasnay flow problem. Various Shape Parameter and nodal

distributions are assessed, and both solvers are used to find results up to Re = 3200

The Modified Method Of Particular Solutions For Solving Navier-StokesSystem Of Equations

This section describes the methods and techniques used in a global MAPS in order to solvean incompressible steady state Navier Stokes system of equations in terms of its primitive

variables (p-). Below in equations 8 and 9, are shown the mass and momentum conservationequations, with i and j representing 1 and 2 for the two dimensional problems described in

this paper.

Equation 8

= 0

Equation 9

= +

2

The natural boundary conditions are defined in terms of their given velocity and/or surface

traction values;


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Equation 10

= , where Equation 11

=

=

+

+

=

, where

To express the velocity and pressure fields in terms of linear superposition of particular

solutions, ( , ), we can define the non-homogenous Stokes system of equations as;Equation 12

Equation 13

, the non-homogenous term in the momentum equation is defined as the MQ RBF or, = (2 + 2)12, this is only dependent on the Euclidean distance r, between the fieldpoint and the trial point . In order to find a close form expression of the particularsolution, Oseens decomposition formula is used to find the corresponding fundamental

solution of the Stokes system, the Stokeslet ( (19) & (20)).

Defining the velocity field () in terms of the auxiliary potential, ensures that thecontinuity equation is already satisfied;

Equation 14

By substituting equation 14 into equation 12, we can find a non-homogenous biharmonic

equation of the potential . Hence, using the RBF as the non-homogenous term, and therelationship between the pressure and the auxiliary potential , as follows;

Equation 15

Equation 16

By integrating equation 15, the resulting expression for is found. The WolframMathematica Online Integrator tool was used to perform the integration (21). All

corresponding singular components of the resulting expression have been omitted, as they are

regarded as complementary solutions of the biharmonic problem.


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Equation 17

The corresponding expressions for the particular solutions velocity and pressure fields arefound by substituting the above expression into the potential into equations 14 and 16. (Theequations of velocity and its derivatives are given in the Appendices)

An important point to observe is that the obtained particular solution is not singular in a

bonded domain . (Again see appendices). Hence the approximated velocity and pressure

fields, , can be found and defined by a linear superposition of N particular solutions at Ntrial points ;

Equation 18

Equation 19

By substituting these equations with the approximation of the directional derivative of the

velocity into a linear version of the momentum equation (equation 9) shown below as;

Equation 20

Using as the guess value of the velocity given by the previous solution of the Picarditeration, we obtain the homogeneous linear superposition of the MQ functions. These

represent the approximate momentum equation;

Equation 21

=

Within equation 21, equation 9 has been used in reference to the last term on the left hand

side, as;

Where =


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Equation 22

In order to find the complete set of equations necessary to complete the collocation process,

either equation 18 or 19 must be substituted into the corresponding boundary condition(equation 10 or 11). Collocating the resulting expression, at either the 1boundary for or2 for , for each of the k=1,2 components, enables the first rows of the matrix system(equation 23) to be found. Further collocation of equation 21 at the internal nodes Ni, for each

of the k=1,2 components allows the last two rows to be obtained, as shown below in equation

23;

Equation 23

The general form of the boundary condition is expressed as 1,2, = () where is the boundary operator and is the corresponding value of the boundary condition. Ifunidirectional flow has already developed, it is necessary to define a zero tangential velocity

at cross sections with prescribed pressure boundary conditions.

As mentioned before, the continuity equation does not need to be explicitly imposed in the

produced system of equations as the formulation is already mass conservative. The usedsuperposition particular solutions also exactly satisfy the continuity equation. It is also

important to note there is no need for a p-v coupling strategy as there is an already inherent

relationship between the respective particular solutions (as shown by equation 16). With

respect to the non-linearity of the Navier-Stokes system of equations, they can be solved by

using a direct Picards iteration method. Used to linearize the momentum equation, a guess at

the velocity field is found from the previous iteration (see equation 20), an initial value isset corresponding to the problem in consideration. The stopping criterion of the Picard

iterations is defined by the L2-norm of the difference between the variable = 1,2, inthe present iteration and the values in the previous one, i.e. 1.

The Simple Gaussian Solver

Within the simple Picard iterative scheme, one must choose a solver for the purpose of

solving a system of linear equations in the form = . In the simplest scheme we can use asimple Gauss Solver, in MATLAB this is defined as mldivide and signified by a simple

backslash. It is the simplest way of solving the system of equations (equation 23) for . Tosolve, the Galois array A is divided into b to produce the particular solution of the linear

equation = . The MATLAB code for this method is shown in lines 78 & 79 of theappendices. It is expected that improvements can be made upon this solver.

The Singular Value Decomposition Solver

SVD (22) & is a well established method of solving a linear system of non-homogeneous

equations. It is expected in this project to provide an improved method with which to solve


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two-dimensional Navier-Stokes systems of equations, within a Picard iteration Scheme. It has

also been indicated to provide more stable results, with the scheme less likely to be sensitive

to changes in the SP value.

The equations to be solved can be considered to be in the form = , SVD allows matrix Ato be decomposed into;

Equation 24

= where the U and V

Tare unitary orthogonal matrices containing the eigenvectors of and

respectively and is the transpose of V. S is a diagonal matrix containing the squareroots of the eigenvectors of U and V

T. Each of the diagonal values of S are the (non-negative)

singular values of A.

To solve for x, the rank of matrix A is found, i.e. the number of singular values of A. The

Pseudoinverse of A, A+

is then found;

Equation 25

+ = +where the diagonal matrix S

+is calculated by finding the reciprocals of the singular values

present within diagonal of S. The solutions to x+

(or in the MAPS case, ) are found using;

Equation 26

+ =+ (= )The MATLAB code for this method is shown in lines 79-84 of Main Code section of the

appendices.

Pre-Estimated Solutions

One way in which the scheme can be made more efficient is with the use of pre-estimated

solutions, for example, as is done in this project, when starting the Picard for a higher

Reynolds number, we can pre insert the previous value of the previous, lower, Reynoldsnumber, that has been previously solved for the same nodal distribution. As mentioned above,

this use is the equivalent of setting an appropriate guess for . This reduces the number ofiterations needed, as the original estimate, is more accurate than when starting from zero. It

also helps produce a more stable result, and in some cases to converge at all. In this project,all results produced for Reynolds numbers over 1000 used this additional process. When

using Reynolds numbers lower than 1000, it is found that solving for both a lower and then a

higher Reynolds number took more time and computing power than just solving from a guess

of zero. It is also noted that there is very little difference between the qualities of results for

each method for = 1000


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Lid Driven Cavity Flow Scenario Numerical Results

Figures 2 and 3 show the sensitivity displayed by the L2 Norm parameter, when the value of

the SP was changed. As we can see, both Solvers display broadly similar characteristics. The

data is based upon the lid driven cavity scenario with = 400 and a nodal distribution of31x31. From these graphs, they look broadly similar, both show the expected characteristic

shape, and numerically they are both within reasonable parameters.

Figure 4: Showing the RMS Error sensitivity with respect to the value of the Shape Parameter

2.00E-04

2.50E-04

3.00E-04

3.50E-04

4.00E-04

4.50E-04

5.00E-04

5.50E-04

6.00E-04

6.50E-04

3.E-04 5.E-04 1.E-03 5.E-03 1.E-021.E-02

RMSE

rror

Shape Parameter Values

SVD Gaussian

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00

L2Norm

Shape Parameter Value

Converging

Non-Converging

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00

L2Norm

Shape Parameter

Converging

Non Converging

Figure 3: Chart to show the sensitivity of L2 Norm

with changes to the value of the Shape Parameter

when usin a Sin le Value Decom osition Solver

Figure 2: Chart to show the sensitivity of L2 Norm

with changes to the value of the Shape Parameter

using a Simple Gaussian Solver


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It is only when we examine Figure 6, above, that we see the differences between the two

solvers. Figure 6 shows a clear advantage in the SVD solver. The RMS error is significantly

smaller for the larger values within the range of Shape Parameter values.

Results were also compared between the Pre-existing Ghia (1) values for 1 & 2, as shownin figures 5 and 6. Again, examination of the graphs above adds to the premise that the SVD

solver provides a more accurate method of solving the system of linear equations resident

within the Picard iteration scheme. Certainly at = 3200 as shown above, the results showonly a very slight improvement in their agreement with the pre-existing Ghia results. The

same has been found for all vales of Re up to 3200. The effect becomes more pronounced as

Re increases, this is a good argument for the implementation of an SVD solver at higher

values of Re.

Figure 5: Showing the obtained u1, u2 velocity profiles upon the lines x1=0.5, x2=0.5, using the SVD solver

Figure 6: Showing the obtained u1, u2 velocity profiles upon the lines x1=0.5, x2=0.5, using the Gaussian solver


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Figure 7 shows the number of iterations needed to converge to within the tolerance set by the

L2 Norm for a range of SP values. Generally for both solvers, the sensitivity to the SP value

was remarkably consistent. It is however worth noting that the SVD approach seemed more

stable when using smaller shape parameters, most notable down to 1105. This in itself isnot particular significant, as better results are achieved with higher Shape Parameter values,

however it does represent a larger range of shape parameter values, which in turn results in

good stability, and hence convergence and good quality results.

Any results above the dotted line represent scenarios where the solutions did not converge,

generally the maximum number of iterations to try was set at 50, and some even exceeded

this number.

Figure 7: Showing the amount of iterations needed to converge on a solution for various Shape Parameters, based

upon a Reynolds number of 400

The Extension To, & Affects of Higher Reynolds Numbers

Figures 8 and 9 show the Flow lines of the lid driven cavity scenario for a Reynolds number

of 3200, and a nodal distribution of 51x51 for each solver. Both Solvers provide very good

results, however as shown below, again the SVD solver provides a more comprehensive

representation of the flow lines at a Reynolds number of 3200. The notoriously under-

resolved areas in the bottom corners are more detailed, as is the centre, and the flow lines

directly under the approach flow of the lid driven cavity flow scenario.

0

5

10

IterationsToConvergance

Shape Parameter

SVD

Gaussian

Did Not

Converge


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Figure 8: Flow Lines Shown For Re=3200

Using a SVD Solver

Both solvers have been used to reach a Reynolds number of 3200, both showing good results

(figures 8 & 9). When reaching these higher Reynolds numbers, they were found using an

estimate for from the solutions for = 1000. The differences in terms of efficiencybetween the two solvers were as expected, with the SVD solver taking slightly longer. For

= 3200 the SVD solver took 19.6 hours, and 14.6 hours for the Gaussian solver (Table 1,Appendix).

It is also important to note that a smaller value of the Shape Parameter had to be used for the

extension to higher Reynolds numbers. This is due to the ill conditioning of the interpolating

matrix that occurs as Re increases. Although the author attempted to reach higher than 3200,i.e. = 5000, time constraints meant it was not possible to pursue this to a great extent,although promising work was done towards this goal. It is confidently predicted that higher

values of Re could be achieved with greater nodal distributions, and points within them, this

is mentioned further in the recommendations section below, but sadly time constraints did not

allow this to be pursued within the scope of the project.

Computing Efficiency

Figure 10: Showing the Time Taken to Converge for various values of the Shape Parameter, using variables of

Re=400 and a 31x31 Nodal Distribution

The computing efficiency of each solver

has also been investigated within the

project, shown to the left, figure 10 shows

the overall time to converge for the

various SP values. As we can see the

Gaussian method seems very slightly

quicker overall for the range of values,

which is as mentioned above. It is worth

mentioning that the multiple computers

used were not being used heedlessly. Due

to the fact that the majority of theuniversity computers were not up to the

job of processing the amount of data,

individual computers had to be used, and

Figure 9: Flow Lines Shown For Re=3200Using A Direct Gauss Solver

0

1

2

3

4

5

1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00TimeTakenToConverge(hrs)

Shape Parameter

Gaussian

SVD


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as many of the scripts took 5+ hours to run, it was impossible to not use them for other work

during that time. Hence some of the processing and memory operations may have been

compromised from their optimum performances when running the scripts.

Conclusions

The aim of the project discussed in this report has brought to the fore a number of

conclusions, based upon the results discussed above. The overall method has as discussed has

not changed to a great extent, but by working with the scheme and corresponding code over

the past 6 months, small tweaks and additions have been added to further analyze and

improve the MATLAB code. Use of both solvers, Gaussian, and SVD, has raised further

possibilities and questions with respect to the MAPS. The SVD solver proved to be more

accurate, and have greater detail, as witnessed by the graphs above. For all converging values

of the SP, it averaged an RMS Error of 3.5395E-04, compared with the average for the

Gaussian Solver of 3.9620E-04, almost a 12% increase in accuracy.

The Gaussian Solver however, proved to take the least time, averaging 10% quicker than the

SVD Solver. As expected, it also took up less computing power, averaging for a relatively

low Reynolds number of 400 and a medium nodal distribution of 31x31, 30Mbs less needed

per iteration. At low Reynolds numbers and coarser nodal distributions then, where

computing power is at a premium, the Gaussian solver is more useful. In the instance where

computing power is not an issue, for instance in many of many now common medium to high

end computers, an SVD solver must be considered, as it provides an increase in both

accuracy and stability.

Recommendations

Other Solverso LU decompositiono Etc.

Higher Re testingo Re=5,000o Re=10,000

Refined Meshes

o 101x101o 161x161

Use of headless operating system, i.e. without the clutter of multiplexing, or otherprocessor or memory demands.

Acknowledgements& References

The current report is part of work currently ongoing by H. Power2, C. A. Bustamante

3and W.

F. Florez2. Thanks are given to Dr. Bustamante for the original version of the MATLAB

4

2 School of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, University Park,

Nottingham, NG7 2RD, UK3Instituto de Energa, Materiales y Medio Ambiente, Universidad Pontificia Bolivariana, Circ. 1 No. 74-34,

Medelln, Colombia4 http://www.mathworks.co.uk/products/matlab/ 4 http://www.mathworks.co.uk/matlabcentral/


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code and a very large thanks to Professor Power for his insights and advice on this project. I

also have to thank the MIT Open Courseware project (23) video for help with solving Linear

equations using SVD. I am also indebted to the good people at MATLAB Central5, whose

forums have been ever helpful when searching for the correct functions or operators to use.

Further thanks are due to Joshua Greenslade, for use of his computer when running high end

scripts.

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D. 2005, Eng. Anal., pp. 343-353.9.Approximation of function and its derivatives using radial basis function networks. Mai-Duy, T. T.-C.

N. 2003, Appl. Math., pp. 197-220.10.Numerical solution of diferential equations using multiquadric radial basis function networks. N. Mai-

Duy, T. Tran-Cong. 2001, Neural Networks 14, pp. 185-199.

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Shu, K. Yeo, D. Xu. 2006, Comput. Methods Appl. Mech. Engrg. 195, pp. 197-220.

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http://integrals.wolfram.com/index.jsp.22. Singular Value Decomposition (SVD) Tutorial.Massachusetts Institute of Technology. [Online]

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Appendices

Figures and Tables

Table 1: Showing some of the numerical results for various Nodal Distributions, and Re and SP valuesModified Method Of Particular Solutions

'Gaussian (mldivide)' 'Single Value Decomposition (SVD)'

Node

Configuration

Reynolds

Number, Re

Shape

Parameter, cConverged? Iterations Time(hrs) L2 Norm Converged? Iterations Time (hrs) L2 Norm

21x21 100 1.00E-02 Y 5 0.1933 - Y 5 - -

41x41 100 1.00E-02 Y 5 5.3722 - - -

51x51 100 1.00E-02 Y 5 6.6408 - - - - -

41x41 400 1.00E-02 Y 5 2.9281 - - - - -

51x51 400 5.00E-03 Y 5 6.5808 - Y 5 - -

51x51 400 5.00E-02 N - - - - - - -

41x41 800 1.00E-02 Y 14 8.1303 - - - - -

51x51 400 2.50E-02 N - - - - - - -

51x51 400 1.25E-02 N - - - - - - -

51x51 400 7.50E-03 Y 6 8.1289 - - - - -

31x31 400 1.00E-06 N(Max) 25 - - - - -

31x31 400 1.00E-05 N (Max) 25 4.2380 8.9013E-01 N(Max) 25 5.1287 -

31x31 400 1.00E-04 N (Max) 25 4.0577 5.2321E-01 N(Max) 25 4.7108 8.9928E-02

31x31 400 1.25E-04 - - - - N(Max) 50 - 7.5300E-02

31x31 400 2.50E-04 0.9213 4.8898E-02 Y 5 0.9708 4.8899E-02

31x31 400 5.00E-04 - - - - Y 5 0.9601 4.9276E-02

31x31 400 1.00E-03 Y 5 0.8845 5.0909E-02 Y 5 0.9416 5.0900E-02

31x31 400 5.00E-03 - - - - Y 5 0.9693 8.7543E-02

31x31 400 1.00E-02 Y 5 0.9321 2.2272E-01 Y 5 1.2166 2.2271E-01

31x31 400 1.10E-02 Y 5 0.8485 - - - -

31x31 400 1.15E-02 Y 5 0.8577 - - - -

31x31 400 1.25E-02 Y 5 - 3.6517E-01 Y 5 0.9801 3.6524E-01

31x31 400 2.50E-02 Y 7 1.1973 Y 7 1.3196 4.8828E+00

31x31 400 3.75E-02 - - - - N(Max) - - 2.6225E+01

31x31 400 5.00E-02 N(Max) 25 4.1050 6.5739E+03 N(Max) 25 4.5353 1.0500E+03

31x31 400 5.00E-02 N (Max) 50 8.3614 - - - - -

31x31 400 1.00E-01 N (Max) 25 4.5099 - - - - -

51x51 1000 1.00E-02 Y 7 - - Y 7 - -

51x51 3200 1.00E-03 Y 11 14.6206 - Y 5 19.6580 -

MATLAB Code

Main Code

Included below is a copy of the main code, functions called are not included, however they

are explained within the existing comments;

1. % Code for the solution of the 2-D laminar flow problems (Lid Driven Cavity Flow) bythe Global Method of

2. % approximated Particular Solutions w/ refined mesh and pressure computation)3. tic; %start time measurement

4. clearall

5. loadnodes.dat %File containing node coordinate6. %load RENS.txt %File containing initial solution

7. x=nodes(:,2:3);8. s1=size(x);9. n=s1(1);10.[bp,ip,n1,n2,bk]=geosor2a(x,n); %bp(:,1:2) contains the coordinates of boundary points

and bp(:,3:4) the velocity values at boundary %ip(:,1:2) contains the coordinates ofinternal points

11.%Global system computing aa*al=bb12.den=400; %Assuming Density (Re=den*L*U/mu, if L=1,U=1,mu=1 then Re=density)13.sp=0.0125; %Shape Parameter (c)

14.aa=zeros(2*(n1+n2),2*(n1+n2));15.bb=zeros(2*(n1+n2),1);

16.tolp=0.01; %Tolerance17.k=1; %initialise count18.nmax=50; %Max Iterations19.up0=zeros(n1+n2,3);20.u1=zeros(n1+n2,n1+n2);


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21.u2=zeros(n1+n2,n1+n2);22.pp=zeros(n1+n2,n1+n2);23.%up0=RENS(:,1:3); %Uncomment when a initial guess different to zero has to be set.

24.while (k


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88.up=zeros(n1+n2,3);89.xp(1:n1,1:2)=bp(1:n1,1:2);90.fori=1:n191.if (bk(i)==0)92.up(i,1)=bb(i,1);93.up(i,2)=bb(i+n1,1);94.up(i,3)=dot(pp(i,:),al(:));95.elseif (bk(i)==1)

96.up(i,1)=dot(u1(i,:),al(:));97.up(i,2)=bb(i+n1,1);98.up(i,3)=bb(i,1);99.end100. end101. up(n1+1:n1+n2,1)=u1(n1+1:n1+n2,:)*al;102. up(n1+1:n1+n2,2)=u2(n1+1:n1+n2,:)*al;103. up(n1+1:n1+n2,3)=pp(n1+1:n1+n2,:)*al;104. tol1(k)=abs(max(up(:,1)-up0(:,1))) %/abs(max(up0(:,1)))

105. %Saving results106. dlmwrite('RENS.txt',up, 'delimiter', '\t', 'precision', 6); %saving RENS data107. dlmwrite('RADNS.txt',xp, 'delimiter', '\t', 'precision', 6); %saving RADNS data

dlmwrite('toles.txt',tol1, 'delimiter', '\t', 'precision', 6); %savingtolerances data

108. if (tol1(k)

Documents

Proving The Solubility Of A Navier-Stokes Solution Using A Modified Method Of Particular Solutions