Constructing Simulations of PDEs

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10.34: Numerical Methods Applied to

Chemical Engineering

Finite Volume Methods

Constructing Simulations of PDEs

1

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Recap

•

von Neumann stability analysis

•

Finite volume methods

2

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3

•

Generally used for conservation equations of the form:

•

is the density of a conserved quantity

•

is the flux density of a conserved quantity

•

The integral version of such an equation is:

Finite Volume Method

@ b

@ t = r · j + r (x, t)

b(x, t) j(x, t)

d dt

Z

V ^⇤

b(x, t) dV = Z

S^⇤

n · j(x, t) dS + Z

r (x, t) dV

V ^⇤

d B

^⇤

(t) = F

^⇤

(t) + R

^⇤

(t)

dt

_ACC IN/OUT GEN/C

or

*

ON

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4

•

What are each of these terms?

• ^B

^⇤

^{(t) =} ^V

^⇤

^¯ ^b

^⇤

^(t)

• ^R

^⇤

^{(t) =} ^V

^⇤

^r ^¯

^⇤

^(t)

• ^F

^⇤

^{(t) =} ^X ^F

^k

^{(t) =}

k2faces^⇤ k2

X A

^⇤_k

(n

_k

•

^faces^⇤

· j)(t)

the sum of fluxes through each face of the volume *

d ¯ b

^⇤

V

^⇤

= F

_k

(t) + V

^⇤

r ¯

^⇤

(t) dt

•

We want to solve for by approximating the reaction and flux terms. Let’s construct low order approximations physically.

d B

^⇤

(t) = F

^⇤

(t) + R

^⇤

(t)

dt

_ACC IN/OUT GEN/CON

*

¯ b(t)

X

k2faces^⇤

Finite Volume Method

Conservation within a finite volume:

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5

*

@ b

@ t = r · j + r(x, t)

d ¯ b

^⇤

V

^⇤

=

dt

k2

X F

_k

(t) + V

^⇤

r ¯

^⇤

(t)

faces^⇤

Finite Volume Method

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6

*

@ t = r · j + r(x, t)

d ¯ b

^⇤

V

^⇤

=

dt

k2

X F

_k

(t) + V

^⇤

r ¯

^⇤

(t)

faces^⇤

Finite Volume Method

@ b

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7

@ b

@ t = r · j + r(x, t)

d ¯ b

^⇤

V

^⇤

= X

F

_k

(t) + V

^⇤

r ¯

^⇤

(t)

, France

dt

k2faces^⇤

*

Geometrica: INRIA

Finite Volume Method

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8

d ¯ b

^⇤

V

^⇤

=

dt

k2

X F

_k

(t) + V

^⇤

r ¯

^⇤

(t)

faces^⇤

Cardiff et al. J Biomech Eng 136(1), 2013

*

Finite Volume Method

@ b

@ t = r · j + r(x, t)

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Numerical Solution of PDEs

L

Step 1: domain decomposition finite difference: nodes

i, j

W

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10

Numerical Solution of PDEs

L W

Step 1: domain decomposition finite volume: cells

i, j

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L

11

W

Numerical Solution of PDEs

Step 1: domain decomposition

finite element: elements (local basis functions)

i, j

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12

Numerical Solution of PDEs

Step 1: domain decomposition

Always choose the spacing between nodes/dimensions of cells to match the physics.

Never pick a certain number of nodes or cells a priori. That number is irrelevant.

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13

19

x-coordinate (cm)

cm)( etanidrocoy-

Concentration profile in (x,y)-space. Thin medical implant at x [1,2] cm.

0 5 10 15 20 25 30

0 0.5 1 1.5 2

-2 0 2 4 6 x 108 ^-4

x-coordinate (cm)

y-coordinate (cm)

Close-up of concentration profile in (x,y)-space.

R = 30, B = 2, N = 300. Semi-infinite boundary condition at x=R and y=B: Dirichlet zero concentration.

0 2 4 6 8 10

0 0.02 0.04 0.06 0.08 0.1

-2 0 2 4 6 x 108 ^-4

Figure 6.2 Concentration profile using Dirichlet boundary condition for x = R and y = B. R = 30 cm, B = 2 cm, and N = 300.

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Numerical Solution of PDEs

Step 2: formulate an equation to be satisfied at each node/cell

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15

Numerical Solution of PDEs

Step 2: formulate an equation to be satisfied at each node/cell Example: at interior node/cell i,j

r

²

c = 0

equation i,j:

c

_i+1,j

+ c

_i _1,j

+ c

_i,j ₁

+ c

_i,j₊₁

4c

_i,j

= 0

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16

Numerical Solution of PDEs

Step 2: formulate an equation to be satisfied at each node/cell Example: at boundary node/cell i,j

equation i,j:

c = 1

c

_i,j

1 = 0

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17

Numerical Solution of PDEs

Step 3: solve the system of equations formulated at each node/cell for the value of unknown function at each node/cell

f (c) = 0

If equations are linear, use linear iterative methods

If equations are nonlinear, use nonlinear iterative methods

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18

Numerical Solution of PDEs

Step 3: solve the system of equations formulated at each node/cell for the value of unknown function at each node/cell

f (c) = 0

must be a vector of the unknowns must be a vector of the equations

c

f

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Numerical Solution of PDEs

1 N

_x

N

_y

1 2

i j

Indexing

k = i + (j 1)N

_x

c

_i,j₊₁

= c

_k+N_x

c

_k

= c

_i,j _or

k = j + (i 1)N

_y

c

_i,j₊₁

= c

_k+1

3,

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20

Numerical Solution of PDEs

y

1 , 2

i j

Indexing

1 N

_x

N

k = i + (j 1)N

_x

f

_k

(c) = f

_i,j

(c)

^or

k = j + (i 1)N

_y

3

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21

Numerical Solution of PDEs

N

_y

1 2

j

Indexing

1 N

_x

k = i + (j 1)N

_x

f

_k

(c) = f

_i,j

(c)

^or

k = j + (i 1)N

_y

3,

i

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22

Numerical Solution of PDEs

N

_x

y

N

_z

N

c

_l

= c

_i,j,k

, l =?

Exercise: write a single index for finite difference nodes in a cubic domain with (Nx, Ny, Nz) nodes in each cartesian direction

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Numerical Solution of PDEs

Exercise: write a single index for finite difference nodes in a cubic domain with (Nx, Ny, Nz) nodes in each cartesian direction

N

_x

N

_y

N

_z

c

_l

= c

_i,j,k

, l = i + (j 1)N

_x

+ (k 1)N

_x

N

_y

23

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Numerical Solution of PDEs

Example: solve the diffusion equation in 2-D on a square with side = 1.

r

²

c = 0 c = 0

c = 0

c = 1

f (c) = 0

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Numerical Solution of PDEs

h = 1 / 10; % Spacing between finite difference nodes Nx = 1 + 1 / h; % Number of nodes in x-direction

Ny = Nx; % Number of nodes in y-direction

c0 = zeros( Nx * Ny, 1 ); % Initial guess for solution

c = fsolve( @( c ) my_func( c, Nx, Ny ), c0 ); % Find root of FD equations

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Numerical Solution of PDEs

function f = my_func( c, Nx, Ny )

% Loop over all nodes for i = 1:Nx

for j = 1:Ny

k = i + ( j - 1 ) * Nx; % Compound index

% Boundary nodes if ( i == 1 )

f( k ) = c( k );

elseif ( i == Nx ) f( k ) = c( k );

elseif( j == 1 )

f( k ) = c( k ) - 1;

elseif( j == Ny ) f( k ) = c( k );

% Interior nodes else

f( k ) = c( k + 1 ) + c( k - 1 ) + c( k - Nx ) + c( k + Nx ) - 4*c( k );

end;

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Numerical Solution of PDEs

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.08 seconds to solve

h = 1/10

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Numerical Solution of PDEs

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

h = 1/100

700 seconds to solve!

Why is it almost 10,000x slower?

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Numerical Solution of PDEs

f (c) = 0 = Ac b

function [ Ac, b ] = my_func( c, Nx, Ny ) Ac = sparse( Nx * Ny, 1 );

b = sparse( Nx * Ny, 1 );

% Loop over all nodes for i = 1:Nx

for j = 1:Ny

k = i + ( j - 1 ) * Nx; % Compound index

% Boundary nodes if ( i == 1 )

Ac( k ) = c( k );

elseif ( i == Nx ) Ac( k ) = c( k );

elseif( j == 1 )

Ac( k ) = c( k );

b( k ) = 1;

elseif( j == Ny ) Ac( k ) = c( k );

% Interior nodes else

Ac( k ) = c( k + 1 ) + c( k - 1 ) + c( k - Nx ) + c( k + Nx ) - 4*c( k );

end;

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Numerical Solution of PDEs

h = 1 / 10; % Spacing between finite difference nodes Nx = 1 + 1 / h; % Number of nodes in x-direction

Ny = Nx; % Number of nodes in y-direction

% Calculate RHS of Ac = b

[ Ac, b ] = my_func( zeros( Nx * Ny, 1 ), Nx, Ny );

% Find solution of linear FD equations using the an iterative method

% This is gmres (generalized minimum residual). Other choices include

% bicgstab (conjugate gradient), minres (minimum residual), etc.

% The requires a function that returns A*c given c.

c = gmres( @( c ) my_func( c, Nx, Ny ), b, 100, 1e-6, 100 );

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Numerical Solution of PDEs

0.015 seconds to solve!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

h = 1/10

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Numerical Solution of PDEs

5 seconds to solve!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

h = 1/100

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Numerical Solution of PDEs

function [ Ac, b ] = my_func( c, Nx, Ny ) Ac = sparse( Nx * Ny, 1 );

b = sparse( Nx * Ny, 1 );

k = i + (j 1)N

_x

% Define indices of boundary points and interior points bottom = [ 1:Nx ];

top = Nx*Ny - [ 1:Nx ];

left = [ 1:Nx:Nx*Ny ];

right = [ Nx:Nx:Nx*Ny ];

interior = setdiff( [ 1:Nx*Ny ], [ left, right, bottom, top ] );

Ac( left ) = c( left );

Ac( right ) = c( right );

Ac( top ) = c( top );

Ac( bottom ) = c( bottom );

b( bottom ) = 1;

A( interior ) = c( interior - 1 ) + c( interior + 1 ) + c( interior - Nx ) + c( interior + Nx ) … - 4 * c( interior );

f (c) = 0 = Ac b

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Numerical Solution of PDEs

1.2 seconds to solve!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

h = 1/100

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10.34 Numerical Methods Applied to Chemical Engineering

Fall 2015

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Constructing Simulations of PDEs

10.34: Numerical Methods Applied to

Chemical Engineering

Finite Volume Methods