## 五年专注**布里斯班论文代写** 信誉保证

turnitin检测 保证原创率 高分通过

本公司成立以来，在**澳洲代写论文**领域获得了不错的口碑，98%以上的客户顺利通过..欢迎大家进行咨询和享受公司为你提供的全方位服务！不论你的英语论文有多难，deadline有多急，我们将给你带来最专业可靠的**澳洲代写论文**服务。

# 澳洲数学assignment代写

SCIENTIFIC COMPUTING -- Due: 3 May 2013

Exercise #3 3 11 March 2013

Find the electric potential ( l/N, m/N, n/N) for the charge distribution:

= -200 3/8 < x < 1/2 , 3/8 < y < 5/8, 7/16 < z < 9/16,

= 0 x=0

= 200 1/2 < x < 5/8 , 3/8 < y < 5/8, 7/16 < z < 9/16,

You may assume that =0 everywhere else in the cube.

Find the solutions jkl for N = 16, 32, 48, 64, 80, 96,128, 160, 192, 256, 320 … until the problem

is too big for the PC.

Suggestions:

a. Allocate type double arrays of size N3 to hold the input, the output and the work arrays.

{I would recommend allocating these three arrays as single dimension arrays of size N3 .

Use the mapping p = j+ Nk + N2l to find the location of vertex j,k,l }

b. Set the jkl array to zero. For j from 3N/8 to N/2-1, for k from 3N/8 to 5N/8 and for

l from 7N/16 to 9N/16 set jkl = -200. Repeat for j from N/2+1 to 5N/8 with jkl = 200.

c. Find the Discrete Fourier transform of this distribution jkl by:

i. For k from 0 to N-1, For l from 0 to N-1, do the size N Discrete Fourier Transform

starting at 0kl with an initial skip of 1.

ii. For j from 0 to N-1, For l from 0 to N-1, do the size N Discrete Fourier Transform

starting at j0l with an initial skip of N.

iii. For j from 0 to N, For k from 0 to N, do the size N Discrete Fourier Transform

starting at jk0 with an initial skip of N2.

d. Calculate mnp from mnp using equation (8.2). 000 should be zero, so set 000 = 0.

e. Calculate the result vertex values of jkl by performing three sets of N^2 of size N Discrete

Fourier Series reversing step c.

You may wish to write 1-D and 2-Discrete Fourier Poisson solvers to get the details correct first.

Partial credit will be given for working 1-D and 2-D codes. Use the given 3-D charge distribution to

define the 2 D and 1-D charge distributions.

As the charge distribution is purely real and the resultant electric potential field is purely real, the

half of the calculation using imaginary values is wasted. It is possible to calculate the Discrete

Fourier Transform of Real Data efficiently when you are doing multiple transforms of the same size

by putting one set of real data into the real part of a size N complex vector and putting another set

of real data into the imaginary part. The Discrete Fourier Transform y of a purely real Data vector x

has the property: yN-k = yk* , the reversed second half of the transform array is the complex

conjugate of the first half. For a purely imaginary Data vector x, the y vector has the property:

yN-k = -yk* .

How long do steps c, d & e take for N = 16, 32, 48, 64, 80…. etc. ?

For what values of N does available computer memory become a problem?

In real modelling of atoms, molecules and semiconductors by this method , the distribution of

charges is averaged onto the nearest jkl grid vertex, the resultant electric potential at each grid

point jkl is calculated by the above algorithm and then each charge is moved by Maxwell’s laws in

the resultant Electric field. This process is repeated several thousand times for each model!

The results for the potential of our dipole brick may converge much faster if we set the faces to +

100, the edges to + 50 and the corners to + 25 (do you have any ideas why!). Test this for the 3-D

case and for any 2-D and 1-D cases if you have coded those up.

Plot j(N/2)(N/2), (N/2)k(N/2) & (N/2)(N/2)l for 0 < j,k,l < N on the same scale for your three largest

values of N. How long does this method of solving Poisson’s Equation take for N = (3,4,5) 24+k

0 < k < (as large as possible).? How would you expect the time taken to increase as N increases

(to leading order as a power of N!). For what values of N does available computer memory become

a problem?