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Even though we use a simple but ridiculous problem finding the optimum saury baking layout on a f...

Even though we use a simple but ridiculous problem finding the optimum saury baking layout on a fish gridiron by Joule heat, we can invoke the interest to science by combining electrical engineering, linear algebra and probability viewpoints. These elements are, use of solving linear equation and Poisson’s equation, and applying the central limit theorem to this situation. In addition, by removing the constraints, we can create a new problem free from our common sense. Presenting funny but essential problems could be another aspect for active learning using the problem of the interdisciplinary scientific methods.

- Hideo Hirose

Hiroshima Institute of Technology

Hiroshima Japan

An Active Learning Method Using the

Problem of Optimum Saury Layout on a

Gridiron by Using the Interdisciplinary

Scientiﬁc Methods

copyright Hideo Hirose

ID 90310 - Network Equation Solving via Node Equation

Network and Poisson Equation

Central Limit Theorem and Network Equation

Central Limit Theorem and Galton Board

Optimum Saury Layout on a Gridiron

In the Case of Water Flow

copyright Hideo Hirose - Raising A Funny but Interesting Problem：

Find the optimum saury baking layout on a ﬁsh gridiron using Joule

heat under the condition that electricity is poured from one edge of

the gridiron and extracted to the opposite side edge.

electricity

primary objective is to teach a method to solve the

systems of linear equation via the node equation method

copyright Hideo Hirose - A

B

C

electricity

Examples for the Answer：

Let’s imagine three kinds of layout.

The answers from students are amazing: almost evenly divided into

two: B and C. This tendency is seen even in researchers and

teachers.

The problem is worth solving because it is not trivial and it includes a

variety of scientiﬁc aspects.

electricity

A

B

C

electricity

electricityWhy they select C?

copyright Hideo Hirose - Network Equation Solving via Node Equation

copyright Hideo Hirose - Network Equation Solving：

There are two types of Kirchhoff’s laws: KVL (Kirchhoff’s voltage

law), and KCL (Kirchhoff ’s current law), and the former is often

introduced in elementary textbooks for electrical engineering and

sometimes in linear algebra. Although the method is easily

understood in the case of toy problems, it becomes difﬁcult to

construct and solve the network equations by using the KVL.

common but not versatile not common but versatile

unknown is current unknown is voltage

loop equation node equation

KVL

KCL

from Wiki

from Wiki - Network Equation via Node Equation：

Node equation method by using the KCL (Kirchhoff ’s current law),

which make the solving algorithm versatile and ease of extension to

very large scale of networks.

copyright Hideo Hirose - 1A

1A

0V

%1V %0.5V

%0.5V

0.5A

0.5A 0.5A

0.5A

A simple circuit (one mesh gridiron)：

1A injection from the left top point

makes a system of linear equations

The equations cannot be solved because of

singular; the redundant equations. Thus, we

force to set x4=0, the equations become,

The solution of x1=-1, x2=-1/2, x4= -1/2 are

obtained.

1A 1Ω

1Ω

1Ω

1Ω

1A

x2

x3

x4

x1

one mesh gridiron

copyright Hideo Hirose - A more complex circuit (four meshes gridiron)：

1A injection to a four meshed gridiron

makes a system of linear equations.

We can solve the equations such that

Since this method can be extended systematically,

we may solve a large scale of gridiron problem.

1A

1A

1A

1A

#1.5V #0.75V#1V

#1V #0.5V#0.75V

#0.75V 0V#0.5V

0.5 0.25

0.25

0.25

0.25

0.25 0.25

0.5

0.5

0.5 0.25

0.25

copyright Hideo Hirose - A much more complex circuit (nine meshes gridiron)：

1A injection to a nine meshed gridiron

makes a system of linear equations

We solve the linear equation Ax=b, and

are obtained.

1"

0"

0"

0"

0"

0"

0"

0"

0"

0"

0"

0"

0"

0"

0"

1 2 3 4 5 6 7 8 9 2 3 4 5

1

2

3

4

5

6

7

8

9

"

2

3

4

5

1

2

3

4

5

6

7

8

9

"

2

3

4

5

1 2 3 4 5 6 7 8 9 2 3 4 5

!

節点の電位分布

1A

1A

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

!13/7 !19/14

!19/14

!15/14

!15/14

)!8/7

!13/14

!13/14

!13/14

!13/14

!11/14

!11/14

!1/2

!1/2

)!5/7

0

1A

voltage distribution is obtained

copyright Hideo Hirose - Current, Voltage, Joule heat Distribution (four meshed gridiron)：

From the node voltages, a current from node to node is calculated.

Then, the current distribution can be obtained.

To understand the current distribution easily,

we assign colors to each node, as well as the

node voltages. To heat distribution colors are

assigned to each mesh.

Then, Joule heat distribution can be computed.

current distribution

!13/7V !19/14

!19/14

!15/14

!15/14

*!8/7

!13/14

!13/14

!13/14

!13/14

!11/14

!11/14

!1/2

!1/2

*!5/7

0V

1A

1A

1A

0.43A

0.29A 0.14A

0.36A 0.29A

0.5A

0.5A

0.29A

0.14A

0.36A 0.43A 0.5A

0.5A0.29A 1A

voltage distribution Joule heat distribution

voltagescurrent Joule heat

copyright Hideo Hirose - 1A

1A

A much more complex circuit (100 meshed gridiron)：

In the case of 100 meshed gridiron, current distribution, voltage

distribution and Joule heat distributions are similarly obtained.

However, we cannot ﬁnd an optimum solution of the saury layout

1A

1A

1A

1A

current distribution voltage distribution Joule heat distribution

voltagescurrent Joule heat

copyright Hideo Hirose - 1A

1A

A further much more complex circuit (900 meshed gridiron)：

In the case of 900 meshed gridiron, current distribution, voltage

distribution and Joule heat distributions are similarly obtained.

Yet, we still cannot ﬁnd an idea for the optimum saury layout.

1A

1A

1A

1A

current distribution voltage distribution Joule heat distribution

voltagescurrent Joule heat

copyright Hideo Hirose - Optimum layout result：

The optimum layout of the saury on the gridiron cannot be

determined clearly. That is, selection of A, B, or C is much the same

for nice baking. Such an answer is unpredictable, and we can now

understand that the students’ answers were divided evenly into two

(B and C).

electricity

A

B

C

electricity

electricity

which one?

unknown

copyright Hideo Hirose - Network and Poisson Equation
- 1A

1A

Inﬁnity meshes gridiron case：

When mesh size is becoming larger and lager in the gridiron, then

the discrete mesh network may be changed to a continuous plane

(surface) where current can ﬂow to any direction.

Then, a system of linear equations will become to a Poisson equation

for voltage distribution.1A

1A

1A

1A

voltage distribution (100 meshes) voltage distribution (900 meshes) voltage distribution (inﬁnity meshes)

This voltage contour

is a result using the

ﬁnite element

method.

In this case also, we cannot ﬁnd the optimum solution of the layout.

voltage

100 mesh 900 mesh

voltage voltage

copyright Hideo Hirose - Water Flow
- Water ﬂow：

Intuitively, we may imagine that the current ﬂow pattern is similar to

the water ﬂow pattern when the water splits into two ditches at each

joint node. Is this correct?

1/2 1/2

1

1/4 1/41/4 1/4

1/8 2/81/8 2/8

1/16 3/16

1/16 3/16

1/8 1/8

3/16 1/16

3/16 1/16

1

1 1

1

1

1

1

1

1

2

3 3

4 46

Water ﬂow

Pascal’s triangle law

How is the ﬂow distribution here?

ratio of ﬂows at each node

copyright Hideo Hirose - Water ﬂow：

No, the ﬂow distribution is different from the pattern shown

previously by solving the system of linear equations. This is because

the current ﬂow does not simply follow the binary splitting law; i.e.,

water ﬂow splits evenly to two ditches from upward to downward.

Water ﬂow

circuit

copyright Hideo Hirose - Central Limit Theorem and Network Equation

copyright Hideo Hirose - 1A

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

1Ω

0.5Ω

0.5Ω

1Ω 1Ω

1Ω 1Ω

1Ω 1Ω

0Ω

0.5Ω 0.5Ω

0Ω1Ω

1Ω

1Ω 1Ω 1Ω

1Ω 1Ω

1Ω 1Ω 1Ω

0Ω

0Ω 0Ω 0Ω

0Ω 0Ω 0Ω

1Ω 1Ω 1Ω

Water ﬂow design：

To design the current ﬂow similar to the water ﬂow, we have to

assign appropriate resistance values to each branch. In the ﬁgure,

slanting lines are seen, which means that in these branches no

currents are observed regardless whether the resistance values are

0Ω or not; resistances of gray lines in row and column are 1Ω, and

those of solid black lines are 0.5Ω. These values are inversely

computed so that the current values ﬂowing into each node are

equivalent to binomial coefﬁcients.

mimic the water ﬂow

by rearranging

the resistance values

copyright Hideo Hirose - Water ﬂow result：

This shows an example for mesh size is equal to 9.

Slanting lines mean that in these branches no currents are observed

regardless whether the resistance values are 0Ω or not; resistances

of gray lines in row and column are 1Ω, and those of solid black

lines are 0.5Ω.

These values are inversely computed so that the current values

ﬂowing into each node are equivalent to binomial coefﬁcients.

1A

1A

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

1Ω

0.5Ω

0.5Ω

1Ω 1Ω

1Ω 1Ω

1Ω 1Ω

0Ω

0.5Ω 0.5Ω

0Ω1Ω

1Ω

1Ω 1Ω 1Ω

1Ω 1Ω

1Ω 1Ω 1Ω

0Ω

0Ω 0Ω 0Ω

0Ω 0Ω 0Ω

1Ω 1Ω 1Ω

1A

1A

#1.75 #1.25 #1.00 #0.875

#0.875

#0.875

#0.875

#1.00

#1.00

#1.25 #0.75

#0.75

#0.75

#0.5*

#0.5* 0.0*

0.5 0.25 0.125

0.250.5 0.125 0.125

0.25 0.25 0.125

0.250.25 0.25 0.25

0.125 0.25 0.25

0.1250.125 0.25 0.5

0.125 0.25 0.5

copyright Hideo Hirose - 1A

1A

0.5 0.25 0.125

0.250.5 0.125 0.125

0.25 0.25 0.125

0.250.25 0.25 0.25

0.125 0.25 0.25

0.1250.125 0.25 0.5

0.125 0.25 0.5

1

1

1

1

2

1

1

1

1

2

1

1

1

1

3

3

Water ﬂow result：

These ﬁgures show distributions for current, voltage, and heat, in the

cases of the number of nodes 16.

current distribution voltage distribution Joule heat distribution Pascal’s triangle law is seen

current, voltage, Joule heat distribution

by rearranged resistance values

copyright Hideo Hirose - 1A

1A

Water ﬂow result：

These pictures show much ﬁner distributions for current, voltage, and

heat, in the cases of the number of nodes 100.

1A

1A

1A

1A

current distribution voltage distribution Joule heat distribution

voltagescurrent Joule heat

similar to water ﬂow

copyright Hideo Hirose - 1A

1A

1A

1A

1A

1A

current distribution voltage distribution Joule heat distribution

Water ﬂow result：

These pictures show much further ﬁner distributions for current,

voltage, and heat, in the cases of the number of nodes 900.

voltagescurrent Joule heat

similar to water ﬂow

copyright Hideo Hirose - Optimum layout result：

Although the current bandwidth becomes narrower than that in the

case that resistance values are equally assigned, the optimum layout

of the saury on the gridiron cannot be determined clearly. That is,

selection of A, B, or C is much the same for nice baking. Such an

answer is unpredictable, and we can now understand that the

students’ answers were divided evenly into two (B and C).

electricity

A

B

C

electricity

electricity

which one?

unknown

copyright Hideo Hirose - Central Limit Theorem and Water Flow

copyright Hideo Hirose - From Binomial Distribution to Normal Distribution：

To mimic the water ﬂow in a network circuit, we have to design the

element resistance values just ﬁt to the water ﬂow.

If element resistance values are arranged in such a manner, we can

see the current ﬂow shown below. This is an example for binomial

distribution expression. Imagine that a binomial

distribution will become a normal distribution

as the ﬂow step is going to inﬁnity.

This is the case of the central limit theorem.

Assume that Xi be 1, 0 with probability p,

q=1-p, and

Sn=X1+X2+...+Xn

Then, Sn can be approximately regarded as a

normal distribution with mean np, and

variance npq. Its standard deviation σ is √npq.

Now, p=q=1/2.

When, n=10, √npq=1.5, Sn moves from 2 to 8

with 95% probability. copyright Hideo Hirose - n=10 n=30 n=100

n=10

n=10

n=30

n=30

n=100n=10

n=1 n=1 n=1

Binomial Distribution goes to Normal Distribution：

When mesh size becomes larger, thin current ﬂow can be observed,

which is much narrower than we imagine. These are the cases of

n=10, n=30, n=100.

copyright Hideo Hirose - Central Limit Theorem and Galton Board

copyright Hideo Hirose - 2× (n×(1/2)×(1/2))

= n

2 n/(2n)= 1/ n

n

n2

Conﬁdence Interval and Galton Board：

A binomial distribution will become a normal distribution.

Then, the 95% conﬁdence interval can be easily obtained such that

One of the experimental

equipments is the

Galton Board.

When n=100, the 95% conﬁdence

interval is approximately 20.

95% copyright Hideo Hirose - 80 steps Galton Board copyright Hideo Hirose
- Relaxing Current Injection and Extraction

Points

copyright Hideo Hirose - Relaxing the Current Injection and Extraction Points：

We may assume that the heat hotspots are located at the current

injection and extraction points.

electricity

copyright Hideo Hirose - 1A

1A

1A

1A

1A

1A

current distribution voltage distribution Joule heat distribution

Relaxing the Current Injection and Extraction Points：

We performed similar computations to before, but the optimum

answer is still unveiled. Only the injection and extraction points are

heated.

copyright Hideo Hirose - 1A

1A

1A

1A

1A

1A

current distribution voltage distribution Joule heat distribution

Relaxing the Current Injection and Extraction Points：

Although the mesh size is becoming larger, we see that still only the

injection and extraction points are heated.

copyright Hideo Hirose - Only the Injection and Extraction points are Heated：

It seems ridiculous to think such a layout; however, this is seen in real

situations (Lake Biwa in Japan).

electricity

no kidding, its real

from the internet - Relaxing the Current Injection and Extraction Points：

Let’s consider the saury layout on the narrow shaped gridiron.

electricity

copyright Hideo Hirose - 1A

1A

1A

1A

1A

1A

節点毎の流入・流出の電流 節点毎の電位分布 メッシュ毎のジュール熱分布

copyright HH

current distribution voltage distribution Joule heat distribution

Relaxing the Current Injection and Extraction Points：

Pictures show distributions for current, voltage, and heat. - 1A

1A

1A

1A

1A

1A

メッシュ毎のジュール熱分布

（抵抗比＝1:10）

メッシュ毎のジュール熱分布

（抵抗比＝1:100）

メッシュ毎のジュール熱分布

（抵抗比＝1:100000）

Joule heat distribution

resistance ratio = 1:10

Joule heat distribution

resistance ratio = 1:100

Joule heat distribution

resistance ratio = 1:100000

Joule heat distribution comparison：

Joule heat distribution comparison when the resistances outside the

narrow region are 10Ω on the left, 100Ω in the middle, and

100,000Ω on the right.

resistance = 10Ω

resistance = 1Ω

resistance = 100Ω

resistance = 1Ω

resistance = 100000Ω

resistance = 1Ω

copyright Hideo Hirose - Concluding Remarks：

Even though we used a simple but ridiculous problem ﬁnding the

optimum saury baking layout on a ﬁsh gridiron by Joule heat, we

could invoke the interest to science by combining electrical

engineering, linear algebra and statistics viewpoints. These

elements were, use of solving linear equation and Poisson's

equation, and applying the central limit theorem to this situation. In

addition, by removing the constraints, we could create a new

problem free from our common sense. In this paper, we have shown

a problem and its solutions that seem simple and easy to solve, but

it is actually difﬁcult to solve and requires many fundamental

knowledge of science.

Presenting funny, but essential problems could be another aspect of

active learning.

copyright Hideo Hirose - Hideo Hirose

Hiroshima Institute of Technology

Hiroshima Japan

thank you

copyright Hideo Hirose