Elliptic Equations Laplace

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Computational Fluid Dynamics - Prof. V. Esfahanian

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Chapter 6

Elliptic EquationsFinite-Difference Method

First Session Contents:

1) Elliptic Equations

2) Dirichlet & Neumann Boundary Conditions

3) Iterative Methods (Jacobi, Gauss-Seidel, SOR)

4) Block Iterative Methods

5) ADI Method

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Laplace equation is a good example for elliptic equations

Ideal fluid flow

Magnetic potential field

Electromagnetic potential field

Stress-strain analysis in solid elasticity

Applications

Elliptic Equations

3

Laplace equation is a good example for elliptic equations

Ideal fluid flow

Magnetic potential field

Electromagnetic potential field

Stress-strain analysis in solid elasticity

Applications

Non-homogenous form of the Laplace equation is called Poison Equation

Sink

Source

Elliptic Equations

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Fourier Law

is the boundary of domain

is the boundary of sub-domain

If there is no sink or source in Ω,

the net heat flux passing ω should be zero

Elliptic Equations

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5

Divergence Theorem

If is continuous in Ω, it should be zero:

k = const.

Elliptic Equations

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Consider a source term in Ω

<0 Sink

>0 Source

Using Divergence theorem we have:

k = const.

Elliptic Equations

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Elliptic Equations

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Using central difference we have

Using uniform mesh and operator theory we have

Dirichlet Boundary Condition

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Hint:

When fm,n is zero (Laplace equation),

um,n is equal to the average of neighbor values

Considering the above equation for all grid points (MxN),

a set of algebraic equations is achieved as follows


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A is a square matrix of order MxN

U and b are a column vector of order MxN


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For dirichlet boundary condition,

internal grid points are

Example

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Example

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Example

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Comments:

Matrices U and b have the same order

Matrix b is affected by boundary conditions and source terms

Matrix A is square and has 5 non-zero elements in each row

Matrix A is called Bounded matrix

Matrix A is a block tridiagonal matrix (number of diagonal blocks

are equal to grid points in y direction and order of each block is

equal to grid points in x direction)

AU=b has a unique solution because A does not have any

zero eigenvalue.

Example

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Neumann Boundary Condition

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Ghost point


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In the above equation, T.E. is O(h2)

Using central difference method for the boundary conditions,

the overall T.E. is O(h2)

To discretize Newman boundary conditions, using central

difference method, we need the ghost points.


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Hint: Matrix A is Singular. The sum of values in each row is equal to zero

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Compatibility Condition

The net heat flux is zero

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The net heat flux is zero

If u is a solution then u+c is also a solution

If U is a solution then U+cI is also a solution

Compatibility Condition

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Element boundary

Volume boundary

Example

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Example

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Example

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Compatibility Equation

Example

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Iterative Methods

Tx=b

Using numerical methods, the governing PDEs convert to system of algebraic

equations as follow:

Using Gauss-Jordan elimination, the above system of equations can be solved.

In difference numerical methods, the rank of matrix T is equal to the grid points

In 3-D problems, for a 20x20x20 grid, the rank of matrix T is 8000. Therefore,

the traditional elimination methods are not efficient and iterative methods can be

used due to lower computational cost and higher efficiency. 28

Laplace Equation

Iterative Methods

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In a computer program 𝑗 → ∞ is not defined

To show the effect of variation in each iteration we have

Iterative Methods

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For each 𝛼 and 𝛽

The minimum rate of convergence occurs when 𝑢 is closed to 1

Speed of convergence

Jacobi Iteration Method

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For large value of N, rate of convergence is low

Suppose that

𝑁 ≫ 1 →𝜋

𝑁≪ 1

Number of iterations ∝ 𝑁2

Jacobi Iteration Method

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Guass-Seidel

Jacobi

Guass-Seidel Iteration Method

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≈ 1≈1

2

For a square


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≈ 1≈1

2

For a square

Gauss-Seidel

Jacobi


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The gauss-Seidel iteration method can be written as

variation of two iteration

The experiments show that the speed of

convergence can be increased by

reducing/increasing this value

Successive Over Relaxation (SOR)

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The Gauss-Seidel method with relation factor is called SOR method

The above equation can be written as follow


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Shows the rate of convergence


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≈ 1≈𝜔

2

For a square

The value of is minimum when


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for a square we have

Gauss-Seidel

Jacobi

SOR

≈ 𝑁2

≈ 𝑁2

≈ 𝑁

There is not a general way to obtain 𝝎


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Example

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Jacobi Method

Gauss-Seidel Method

SOR

Convergence criterion

Example

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Grid points

Grid points

Example

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Iteration Iteration Method

Jacobi

Gauss-Seidel

SORfor for

Example

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Grid points Grid points

Example

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Jacobi

Method

Example

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Grid points

Example

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Block Iterative Methods

All the above equations can be solved by

Tri-Diagonal Matrix Algorithm (TDMA)

Jacobi Method

Gauss-Seidel Method

SOR

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ADI MethodConsider the 2-D unsteady heat equation

Using Crank-Nicolson method we have:

The above equation has 5 unknowns

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In this case, the

coefficient matrix

is pentadiagonal

The Thomas

algorithm (TDMA)

can not be used

There are some

numerical methods to

solve the above problem

which are not as efficient

as TDMA

Using ADI method,

we can use the

TDMA for 2-D

problems

ADI Method

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Step 1 t → ∆t

2

𝜕2𝑇

𝜕𝑥2

𝜕2𝑇

𝜕𝑦2

is discretized implicitly

is discretized explicitly

ADI Method

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ADI Method

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Step 2 t + ∆t

2→ t+∆t

𝜕2𝑇

𝜕𝑥2

𝜕2𝑇

𝜕𝑦2

is discretized explicitly

is discretized implicitly

ADI Method

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ADI Method

𝑛 +1

2

𝑛 + 1

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1)

2)

1) 2)

ADI Method (Amplification Factor)

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Elliptic Equations Laplace

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