اﻧﺘﮕﺮال ﻋﺪدي

(1)

Numerical Integration

Hasan Ghasemzadeh

یدﺪ ی شور ﮏﯿﺎﮑﻣﻮژ

http://wp.kntu.ac.ir/ghasemzadeh

ﯽ ﻮ ﻦﺪ اﺮ ﻪ اﻮ ﯽ هﺎﮕﺸاد

یدﺪ یﺮ لا ﺘا

يدﺪﻋ لاﺮﮕﺘﻧا

يﺮﺳ ﻚﻳ ﻪﺒﺳﺎﺤﻣ ﻪﺑ لاﺮﮕﺘﻧا راﺪﻘﻣ ﻪﺒﺳﺎﺤﻣ ﻞﻳﺪﺒﺗ

Dr. Hasan Ghasemzadeh 2

(2)

يدﺪﻋ لاﺮﮕﺘﻧا

يدﺪﻋ شور ﻲﻣ

ﻣ يﺮﺳ تﻼﻤﺟ زا هدﺎﻔﺘﺳا ﺎﺑ ﺎﻫ لاﺮﮕﺘﻧا ياﺮﺑ ﺪﻧاﻮﺗ ﻪﻃﻮﺑﺮ

و ندﻮﻤﻧ بﺮﺿ ﻚﻳ رد نآ

ﻪﻧزو ﺐﻳﺮﺿ ﺪﻳآ ﺖﺳﺪﺑ

.

ﭙﺳ و يﺮﺳ ﻪﺑ اﺪﺘﺑا رد نآ ﻞﻳﺪﺒﺗ لاﺮﮕﺘﻧا ﻚﻳ يدﺪﻋ ﻞﺣ شور ﺲ

ﺖﺳا زﺎﻴﻧ درﻮﻣ ﺪﺣ ﺎﺗ ﻪﻋﻮﻤﺠﻣ ﻪﺒﺳﺎﺤﻣ زا ﻪﻠﺻﺎﻓ رد دوﺪﺤﻣ ءاﺰﺟا شور رد يدﺪﻋ يﺎﻬﻟاﺮﮕﺘﻧا ﻪﻨﻣاد ﺎﺒﻟﺎﻏ -1

+1 ﺎﺗ ﻲﻣ ﺪﺷﺎﺑ .

ﻲﻣ ﺮﻈﻧ رد ﻪﻧﻮﻤﻧ طﺎﻘﻧ يﺪﻌﺑ ﻚﻳ يﺎﻀﻓ رد ﺪﻧﻮﺷ

.

ﻪﺟرد زا يا ﻪﻠﻤﺟ ﺪﻨﭼ ﻚﻳ ﺐﻳﺮﻘﺗ ﻊﺑﺎﺗ ﺑاﺮﺑ ﺎﺒﻳﺮﻘﺗ و ﻪﻛ دﻮﺷ ﻲﻣ ﻪﺘﻓﺮﮔ ﺮﻈﻧ رد n

ﺮ

،ﺖﺳا ﺎﻨﺒﻣ ﻪﻄﻘﻧ ﺮﻫ رد

لﻮﻬﺠﻣ ﺐﻳاﺮﺿ ﻲﻣ ار

تﻻدﺎﻌﻣ زا ناﻮﺗ

دروآ ﺖﺳﺪﺑ ﺮﻳز :

) ﺪﻧﺮﺑاﺮﺑ تﻻﻮﻬﺠﻣ و تﻻدﺎﻌﻣ داﺪﻌﺗ (.

لاﺮﮕﺘﻧا راﺪﻘﻣ ﺖﺳا ﺮﻳز ترﻮﺼﺑ

:

(3)

ﻪﻛ ددﺮﮔ ﻲﻣ بﺎﺨﺘﻧا يا ﻪﻧﻮﮔ ﻪﺑ يﺮﺳ تﻼﻤﺟ داﺪﻌﺗ لﻮﻬﺠﻣ ﺐﻳاﺮﺿ داﺪﻌﺗ

ﺗ ﺎ

ﺮﺑاﺮﺑ ﺎﻨﺒﻣ طﺎﻘﻧ داﺪﻌﺗ ﻊﺑﺎﺗ ﺎﻬﻧآ رد ﻪﻛ

ار تﻻدﺎﻌﻣ ناﻮﺘﺑ ﺎﺗ ،ﺪﺷﺎﺑ ﺖﺳا مﻮﻠﻌﻣ

دﻮﻤﻧ ﻞﺣ . هدﺎﺳ ﺖﻟﺎﺣ ﻦﻳا رد ﺮﻳز ﻞﻜﺷ رد لﺎﺜﻣ ياﺮﺑ يا ﻪﻘﻧزوذ ﺐﻳﺮﻘﺗ نﻮﻧﺎﻗ

ﮔ رﺎﻜﺑ ﻪﺘﻓﺮ

ﻪﻜﻳرﻮﻄﺑ ،ﺖﺳا هﺪﺷ

ﺖﺳا :

يا ﻪﻘﻧزوذ ﺐﻳﺮﻘﺗ رد ﺎﻨﺒﻣ طﺎﻘﻧ رد ﺮﻳدﺎﻘﻣ و ﻪﻧزو ﺐﻳاﺮﺿ

د ﻲﻄﺧ ﻊﺑﺎﺗ تاﺮﻴﻴﻐﺗ ﺎﻨﺒﻣ ﻪﻄﻘﻧ ود ﻦﻴﺑ ﺐﻳﺮﻘﺗ ﻦﻳا سﺎﺳاﺮﺑ ﻪﺘﻓﺮﮔ ﺮﻈﻧ ر

و هﺪﺷ ﺎﻨﺒﻣ طﺎﻘﻧ رد ﻊﺑﺎﺗ راﺪﻘﻣ ﻊﺑﺎﺗ ﺮﻳدﺎﻘﻣ ﺮﺑاﺮﺑ ﺎﻘﻴﻗد

ﻲﻣ ﺪﻨﺷﺎﺑ .

ﻂﻳاﺮﺷ رد هﺪﺷ ﻪﺋارا ﻲﻣﻮﻤﻋ ﻪﻄﺑار قﻮﻓ ﻪﻄﺑار n = 1

ﺖﺳا .

ﻪﻜﻴﺗرﻮﺻ رد ﺪﺷﺎﺑ دﺮﻓ يدﺪﻋ n

. ﻪﺟرد ﺎﺗ يا ﻪﻠﻤﺟ ﺪﻨﭼ درﻮﻣ رد

يﺎﻄﺧ ﺎﺑ شور ﻦﻳا n

ﻲﻣ ﻞﺻﺎﺣ ار لاﺮﮕﺘﻧا باﻮﺟ ﻢﻛ رﺎﻴﺴﺑ ﺪﻳﺎﻤﻧ

. ﻪﻜﻴﺗرﻮﺻ رد ﻪﻠﻤﺟ ﺪﻨﭼ ياﺮﺑ ،ﺪﺷﺎﺑ جوز n

ﻪﺟرد ﺎﺗ يا n + 1

ﺪﺷ ﺪﻫاﻮﺧ ﻞﺻﺎﺣ ﻖﻴﻗد ﺎﺘﺒﺴﻧ باﻮﺟ .

ﻪﻜﻴﺗرﻮﺻ رد يوﺎﺴﻣ ﺎﻨﺒﻣ طﺎﻘﻧ ﻞﺻاﻮﻓ

ر ﻦﻳا ،ﺪﻧﻮﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧ رد شو

ﻦﺗﻮﻴﻧ - ] ﺲﺗﻮﻛ

Cotes - Newton

[ ﻲﻣ هﺪﻴﻣﺎﻧ دﻮﺷ

.

ﻞﺣ ﻪﺑ زﺎﻴﻧ شور ﻦﻳا ﻪﻟدﺎﻌﻣ n

دراد لﻮﻬﺠﻣ n .

(4)

سوﺎﮔ شوﺮﺑ يدﺪﻋ لاﺮﮕﺘﻧا

Gaussion – quadrature

ﻊﺑﺎﺗ شور ﻦﻳا رد زا ﺐﻳﺮﻘﺗ ﻪﻠﻤﺟ ﺪﻨﭼ

نآ ﻪﺟرد ه ﺖﺳا p

ﺘﺷﻮﻧ ﺮﻳز ﻞﻜﺸﺑ ﻪﻧزو ﺐﻳاﺮﺿ ﺎﺑ يدﺪﻋ يﺮﺳ ﺐﻟﺎﻗ رد لاﺮﮕﺘﻧا ﺦﺳﺎﭘ ﻲﻣ ﻪ

دﻮﺷ :

ﺳﺎﺤﻣ ﺮﻳز ترﻮﺼﺑ هﺪﺷ ﻪﺋارا ﻪﻄﺑار ،ﻲﻠﻴﻠﺤﺗ ترﻮﺼﺑ ﻪﻜﻴﻟﺎﺣ رد ﻪﺒ

ﻲﻣ دﻮﺷ :

ﻲﻣ تﻻدﺎﻌﻣ يﺮﺳود ﺐﻳاﺮﺿ ﻪﺴﻳﺎﻘﻣ ﺎﺑ ﺖﻓﺮﮔ ﻪﺠﻴﺘﻧ ناﻮﺗ

:

ﺐﻴﺗﺮﺗ ﻦﻳا ﻪﺑ p + 1

ﺮﻳدﺎﻘﻣ ندروآ ﺖﺳﺪﺑ ياﺮﺑ ﻪﻟدﺎﻌﻣ

ﻲﻣ تﻻﻮﻬﺠﻣ ﻪﻛ ﺪﻧا هﺪﺷ ﻞﺻﺎﺣ ﺪﻨﺷﺎﺑ

. ﺘﺴﻳﺎﺑ تﻻدﺎﻌﻣ داﺪﻌﺗ ﺖﺳا ﻢﻠﺴﻣ ﻲﻓﺮﻃ زا ﺮﺑاﺮﺑ ﻲ

ﺪﻨﺷﺎﺑ تﻻﻮﻬﺠﻣ داﺪﻌﺗ ﺎﺑ .

ﻲﻨﻌﻳ :

ﻪﻜﻴﺋﺎﺠﻧآ زا ﻲﻣ ﺖﺳا ﺢﻴﺤﺻ دﺪﻋ ﻚﻳ n

ﻪﺸﻴﻤﻫ ﺖﻔﮔ ناﻮﺗ دﻮﺑ ﺪﻫاﻮﺧ دﺮﻓ دﺪﻋ ﻚﻳ p

.

لﺎﺜﻣ ياﺮﺑ ﻲﻨﻌﻳ

:

(5)

ﻊﺑﺎﺗ ﻦﺘﺷاﺬﮔ رﺎﻴﺘﺧا رد ﺎﺑ شور ﻦﻳا رد

ﻪﺟرد زا داﺪﻌﺗ ، P

n + 1

ﻊﺑﺎﺗ زا ﺎﻨﺒﻣ راﺪﻘﻣ

ﻲﻣ زﺎﻴﻧ درﻮﻣ ﺪﻨﺷﺎﺑ

. تﻻﻮﻬﺠﻣ ﻪﻴﻠﻛ ﻲﺘﻟﺎﺣ ﻦﻴﻨﭼ رد ﻪﻛ

ﻲﻣ ﻪﻧزو ﺐﻳاﺮﺿو ﺎﻨﺒﻣ طﺎﻘﻧ ﺖﻴﻌﻗﻮﻣ ﻪﺑ طﻮﺑﺮﻣ تﺎﻋﻼﻃا ﻞﻣﺎﺷ ﺣ ﺪﻨﺷﺎﺑ

ﻞﺻﺎ

ﺪﻳدﺮﮔ ﺪﻫاﻮﺧ .

ﻪﻜﻴﺗرﻮﺻرد n = 0

ﺪﺷﺎﺑ . ﻲﻣ ﺖﺷﻮﻧ ناﻮﺗ :

ﻦﺘﻓﺮﮔ ﺮﻈﻧ رد ﺎﺑ ًﺎﻤﻠﺴﻣ راﺪﻘﻣ ﻚﻳ

) ﺎﻨﺒﻣ ﻪﻄﻘﻧ ﻚﻳ (

ر باﻮﺟ ﻦﻳﺮﺘﻬﺑ رد ا

رد ﻪﻄﻘﻧ ﻪﻛ ﻲﺘﻟﺎﺣ ﻂﻴﺤﻣ ﻂﺳو

دروآ ﺖﺳﺪﺑ دﺮﻴﮔ راﺮﻗ .

رﻮﺻ ﻦﻳا رد ت

ﻞﻜﺷ ﻖﺑﺎﻄﻣ لوا ﻪﺟرد يا ﻪﻠﻤﺟ ﺪﻨﭼ لاﺮﮕﺘﻧا ﻲﻣ ﻪﺒﺳﺎﺤﻣ

دﻮﺷ . ﻦﻳا رد

ﺖﺳا ﺖﻟﺎﺣ .

ﺪﺷﺎﺑ . ﻲﻣ ﺖﺷﻮﻧ ناﻮﺗ :

ﻲﻣ ﻞﺻﺎﺣ ﻲﻄﺧ ﺮﻴﻏ ﻪﻟدﺎﻌﻣ يﺮﺳ ﻞﺣ ﺎﺑ ناﻮﺗ

دروآ ﺖﺳﺪﺑ ار ﺮﻳز يﺎﻬﺑاﻮﺟ :

لاﺮﮕﺘﻧا ﺐﻴﺗﺮﺗ ﻦﻳا ﻪﺑ يا ﻪﻠﻤﺟ ﺪﻨﭼ ﻚﻤﻛ ﻪﺑ ﻪﺟرد 3 ﻞﻜﺷ ﻖﺑﺎﻄﻣ

ﺖﺳا هﺪﺷ ﻞﺻﺎﺣ .

(6)

ﻪﻜﻴﺗرﻮﺻ رد n = 2

،ﺪﺷﺎﺑ ﺶﺷ ﻪﺒﺳﺎﺤﻣ تﺎﻴﻠﻤﻋ رد ﻪﻟدﺎﻌﻣ

ﺪﺷ ﺪﻫاﻮﺧ ﻞﺻﺎﺣ .

ﻲﻣ ﺮﻳز ترﻮﺼﺑ ﺖﻟﺎﺣ ﻦﻳا رد ﺎﻬﺑاﻮﺟ ﺪﻨﺷﺎﺑ

:

ياﺮﺑ ﻪﺟرد ، n=3

ﺖﺳا ﺮﻳز ﻞﻜﺸﺑ ﻪﺠﻴﺘﻧ و ﻞﺻﺎﺣ ﻪﻟدﺎﻌﻣ ﺖﺸﻫ و ، p=7

:

(7)

لﺎﺜﻣ : اﺪﻴﭘ ﺎﻨﺒﻣ ﻪﻄﻘﻧ ود و ﻚﻳ بﺎﺨﺘﻧا ﺎﺑ ار ﺮﻳز لاﺮﮕﺘﻧا راﺪﻘﻣ ﺪﻴﻨﻛ

:

ﺪﺷﺎﺑ :

راﺪﻘﻣ ﺖﺳا ﺮﻳز ترﻮﺼﺑ قﻮﻓ لاﺮﮕﺘﻧا ﻖﻴﻗد :

لﺎﺜﻣ : اﺪﻴﭘ ﺎﻨﺒﻣ ﻪﻄﻘﻧ ود و ﻚﻳ بﺎﺨﺘﻧا ﺎﺑ ار ﺮﻳز لاﺮﮕﺘﻧا راﺪﻘﻣ ﺪﻴﻨﻛ

:

ﺪﺷﺎﺑ :

ﺖﺳا ﺮﻳز ترﻮﺼﺑ قﻮﻓ لاﺮﮕﺘﻧا ﻖﻴﻗد راﺪﻘﻣ :

(8)

يﺪﻌﺑ ﻪﺳ و ود ﺖﻟﺎﺣ رد يدﺪﻋ لاﺮﮕﺘﻧا

ﻚﻳ ﻪﺒﺳﺎﺤﻣ ﻪﺑ ﺮﺠﻨﻣ ﻪﻟﺄﺴﻣ يﺪﻌﺑ ود ﺖﻟﺎﺣ رد ﺪﺷ ﺪﻫاﻮﺧ هﺮﻴﻐﺘﻣ ود ﻊﺑﺎﺗ ﺎﺑ لاﺮﮕﺘﻧا .

ﻲﻨﻌﻳ :

ﻲﻣ ﻲﮔدﺎﺳ ﻪﺑ ،ﺪﺷﺎﺑ ﻞﻴﻄﺘﺴﻣ ﻚﻳ لاﺮﮕﺘﻧا ﻪﻨﻣاد ﻪﻜﻴﺗرﻮﺻرد ﺗ

شور ناﻮ

د رد و هداد ﻢﻴﻤﻌﺗ ﺪﻣﺎﻌﺘﻣ ﺖﻬﺟ ود رد يﺪﻌﺑ ﻚﻳ ﻪﻟﺄﺴﻣ ﺖﻟﺎﺣ ﻪﺑ ار ﻞﺣ و

ﺖﻬﺟ

دﺮﺑ ﺶﻴﭘ رد ﻪﻟﺄﺴﻣ ﻞﻘﺘﺴﻣ رﻮﻄﺑ .

ﺖﻟﺎﺣ رد ﻦﻳاﺮﺑﺎﻨﺑ

ﻲﻣ ﻲﻣﻮﻤﻋ ﺖﺷﻮﻧ ناﻮﺗ

:

د تاﺮﻴﻴﻐﺗ نﺪﻧﺎﻣ ﺖﺑﺎﺛ و ﺖﻬﺟ رد ﻪﻟﺄﺴﻣ تاﺮﻴﻴﻐﺗ ندﻮﻤﻧ رﻮﻈﻨﻣ ﺎﺑ ﺖﻬﺟ ر

ﻲﻣ

ﺖﺷﻮﻧ ناﻮﺗ

:

(9)

ترﻮﺻ ﻪﺑ ﺎﻨﺒﻣ طﺎﻘﻧ ﺖﻴﻌﻗﻮﻣ ﺮﮔا ﺐﻴﺗﺮﺗ ﻦﻳﺪﺑ

ﻖﻴﻗد ﺖﻴﻌﻗﻮﻣ ،ﺪﻧﻮﺷ هداد نﺎﺸﻧ

ﻲﻣ ﺺﺨﺸﻣ هﺪﺷ بﺎﺨﺘﻧا شور سﺎﺳاﺮﺑ ﺎﻬﻧآ ﺪﻧدﺮﮔ

. ﺎﻜﺑ يا ﻪﻠﻤﺟ ﺪﻨﭼ ﻪﻜﻴﺗرﻮﺻ رد ﻪﺘﻓﺮﮔ ر

تﺎﺟرد زا ﺐﻴﺗﺮﺗ ﻪﺑ ﺪﻌﺑ ود رد هﺪﺷ P

1

2

, ﻲﻣ ،ﺪﻨﺷﺎﺑ P تﻼﻤﺟ ﺎﺑ ار ﺎﻬﻟاﺮﮕﺘﻧا ناﻮﺗ

ﻲﻠﻛ

ﻪﻜﻴﺗرﻮﺻ رد ،

داد راﺮﻗ ﻪﺒﺳﺎﺤﻣ درﻮﻣ ،ﺪﻨﺘﺴﻫ .

ﺑ هﺪﺷ هداد نﺎﺸﻧ ﺞﻳﺎﺘﻧ ﻞﻜﺷ ترﻮﺼﺑ يﺪﻌﺑ ود ءاﺰﺟا ياﺮﺑ سﻮﮔ شور دﺮﺑرﺎﻛ ﺎﺑ ﺖﺳﺪ

ﻲﻣ ﺪﻨﻳآ .

ﺚﻠﺜﻣ ءﺰﺟ :

يﺪﻌﺑ ود ﺖﻟﺎﺣ رد ﻪﺒﺳﺎﺤﻣ ﺖﻬﺟ يورﺮﺑ ﺎﻬﻟاﺮﮕﺘﻧا زا ﺚﻠﺜﻣ ءﺰﺟ ﺢﻄﺳ ناﻮﺗ ﻲﻣ ﺮﻳز لوﺪﺟ دﺮﻛ هدﺎﻔﺘﺳا .

ﺐﻳاﺮﺿ و تﺎﺼﺘﺨﻣ لاﺮﮕﺘﻧا ياﺮﺑ رد ﺚﻠﺜﻣ يدﺪﻋ ﻒﻠﺘﺨﻣ داﺪﻌﺗ

ﺎﻨﺒﻣ طﺎﻘﻧ سوﺎﮔ ﻪﻄﻘﻧ ﻚﻳ ﺎﺑ ﺚﻠﺜﻣ ءﺰﺟ

:

سوﺎﮔ ﻪﻄﻘﻧ ﻪﺳ ﺎﺑ ﺚﻠﺜﻣ ءﺰﺟ :

سوﺎﮔ ﻪﻄﻘﻧ رﺎﻬﭼ ﺎﺑ ﺚﻠﺜﻣ ءﺰﺟ :

سوﺎﮔ ﻪﻄﻘﻧ ﺖﻔﻫ ﺎﺑ ﺚﻠﺜﻣ ءﺰﺟ

:

(10)

يﺪﻌﺑ ﻪﺳ ءاﺰﺟا :

رد يﺪﻌﺑ ﻪﺳ ﺖﻟﺎﺣ ﻂﻴﺤﻣ يورﺮﺑ ﺎﻬﻟاﺮﮕﺘﻧا ﻪﺒﺳﺎﺤﻣ ﺖﻬﺟ

مﺮﻫ

ﻲﻬﺟورﺎﻬﭼ ترﻮﺼﺑ يﺪﻌﺑ ﻪﺳ يﺎﻀﻓ رد

ﻞﺻﺎﺣ ﺮﻳز

ﻲﻣ ددﺮﮔ :

.

ﺐﻳاﺮﺿ و تﺎﺼﺘﺨﻣ ﻲﻬﺟو رﺎﻬﭼ يدﺪﻋ لاﺮﮕﺘﻧا

) ﻮﻠﻬﭘرﺎﻬﭼ مﺮﻫ (

ﺎﻨﺒﻣ طﺎﻘﻧ ﻒﻠﺘﺨﻣ داﺪﻌﺗ رد

سوﺎﮔ ﻪﻄﻘﻧ ﻚﻳ ﺎﺑ ﻲﻬﺟورﺎﻬﭼ مﺮﻫ ءﺰﺟ :

سوﺎﮔ ﻪﻄﻘﻧ رﺎﻬﭼ ﺎﺑ ﻲﻬﺟورﺎﻬﭼ مﺮﻫ ءﺰﺟ :

سوﺎﮔ ﻪﻄﻘﻧ ﺞﻨﭘ ﺎﺑ ﻲﻬﺟورﺎﻬﭼ مﺮﻫ ءﺰﺟ

:

(11)

يدﺪﻋ يﺮﻴﮔ لاﺮﮕﺘﻧا



The mapping approach requires us to be able to evaluate the integrations within the domain (- 1…1) of the functions shown.



Integration can be done analytically by using closed-form formulas from a table of integrals (Nah..)

• Or numerical integration can be performed



Gauss quadrature is the more common form of numerical integration - better suited for

numerical analysis and finite element method.



It evaluated the integral of a function as a sum of a finite number of terms

(12)

يدﺪﻋ يﺮﻴﮔ لاﺮﮕﺘﻧا



W

_i

is the ‘weight’ and 

_i

is the value of f(=i)

يﺮﻴﮔ لاﺮﮕﺘﻧا يدﺪﻋ

- ﺮﭽﻳرداﻮﻛ سوﺎﮔ شور ﻪﺑ



If  is a polynomial function, then n-point Gauss quadrature yields the exact integral if  is of degree 2n- 1 or less.

• The form =c

₁

+c

₂

 is integrated exactly by the one point rule

• The form =c

₁

+c

₂

c

₂



^

is integrated exactly by the two point rule

• And so on…

• Use of an excessive number of points (more than that required) still yields the exact result



If  is not a polynomial, Gauss quadrature yields an approximate result.

• Accuracy improves as more Gauss points are used.

(13)

يدﺪﻋ يﺮﻴﮔ لاﺮﮕﺘﻧا -

سوﺎﮔ شور ﻪﺑ ﺮﭽﻳرداﻮﻛ

- ﺪﻌﺑود ي



In two dimensions, integration is over a

quadrilateral and a Gauss rule of order n uses n

²

points



Where, W

_i

W

_j

is the product of one-dimensional weights. Usually m=n.

• If m = n = 1,  is evaluated at  and =0 and I=4

₁

• For Gauss rule of order 2 - need 2

²

=4 points

• For Gauss rule of order 3 - need 3

²

=9 points

يدﺪﻋ يﺮﻴﮔ لاﺮﮕﺘﻧا -

ﺮﭽﻳرداﻮﻛ سوﺎﮔ شور ﻪﺑ -

ﺪﻌﺑود ي

(14)

رد ﺎﻨﺒﻣ طﺎﻘﻧ داﺪﻌﺗ لاﺮﮕﺘﻧا

يﺮﻴﮔ يدﺪﻋ

 All the isoparametric solid elements are integrated numerically.

Two schemes are offered: “full” integration and “reduced”

integration.

• For the second-order elements Gauss integration is always used because it is efficient and it is especially suited to the

polynomial product interpolations used in these elements.

• For the first-order elements the single-point reduced- integration scheme is based on the “uniform strain

formulation”: the strains are not obtained at the first-order Gauss point but are obtained as the (analytically calculated) average strain over the element volume.

• The uniform strain method, first published by Flanagan and Belytschko (1981), ensures that the first-order reduced- integration elements pass the patch test and attain the accuracy when elements are skewed.

• Alternatively, the “centroidal strain formulation,” which uses 1- point Gauss integration to obtain the strains at the element center, is also available for the 8-node brick elements in ABAQUS/Explicit for improved computational efficiency.

رد ﺎﻨﺒﻣ طﺎﻘﻧ داﺪﻌﺗ يدﺪﻋ يﺮﻴﮔ لاﺮﮕﺘﻧا

The differences between the uniform strain formulation and 

the centroidal strain formulation can be shown as follows:

(15)



Numerical integration is simpler than analytical, but it is not exact. [k] is only approximately integrated

regardless of the number of integration points

• Should we use fewer integration points for quick computation

• Or more integration points to improve the accuracy of calculations.

• Hmm….

(16)

داﺪﻌﺗ ﺶﻫﺎﻛ رد ﺎﻨﺒﻣ طﺎﻘﻧ

ﺮﻴﮔ لاﺮﮕﺘﻧا ي

نآ ﺶﻫﺎﻛ موﺰﻟ و يدﺪﻋ

 A FE model is usually inexact, and usually it errs by being too stiff. Overstiffness is usually made worse by using more Gauss points to integrate element stiffness matrices

because additional points capture more higher order terms in [k]

 These terms resist some deformation modes that lower order tems do not and therefore act to stiffen an element.

 On the other hand, use of too few Gauss points produces an even worse situation known as: instability, spurious

singular mode, mechanics, zero-energy, or hourglass mode.

• Instability occurs if one of more deformation modes happen to display zero strain at all Gauss points.

• If Gauss points sense no strain under a certain deformation mode, the resulting [k] will have no resistance to that deformation mode.

رد ﺎﻨﺒﻣ طﺎﻘﻧ داﺪﻌﺗ ﺶﻫﺎﻛ ﺮﻴﮔ لاﺮﮕﺘﻧا

ي

 Reduced integration usually means that an integration scheme one order less than the full scheme is used to integrate the element's internal forces and stiffness.

• Superficially this appears to be a poor approximation, but it has proved to offer significant advantages.

• For second-order elements in which the isoparametric

coordinate lines remain orthogonal in the physical space, the reduced-integration points have the Barlow point property (Barlow, 1976): the strains are calculated from the

interpolation functions with higher accuracy at these points than anywhere else in the element.

• For first-order elements the uniform strain method yields the exact average strain over the element volume. Not only is this important with respect to the values available for output, it is

(17)

ي

ءﺰﺟ ﻲﻧورد يﺎﻫ ﻲﮕﺘﺴﺑاو زا ﺎﻨﺒﻣ طﺎﻘﻧ ﺶﻫﺎﻛ



) ﺑ ﺎﻳ يﺮﻳﺬﭘﺎﻧرﺎﺸﻓ ﺮﺛا شﺮ

ﻲﺸﻤﺧ ءاﺰﺟا شﺮﺑرد ﻒﺷﺮﻴﻛ (

ﺪﻫﺎﻛ ﻲﻣ .

ﺷ ﻞﻔﻗ هﺪﻳﺪﭘ ﺶﻳاﺪﻴﭘ ﺐﺟﻮﻣ ﺮﺘﻠﻣﺎﻛ يﺮﻴﮕﻟاﺮﮕﺘﻧا ،ءاﺰﺟا ﻪﻧﻮﮔ ﻦﻳارد



و ﻲﮔﺪ

اﺪﻌﺗ ﺶﻫﺎﻛ ﺎﺑ ﻪﻛ هﺪﻳدﺮﮔ ﺖﻴﻌﻗاو زا رود ﺞﻳﺎﺘﻧ و ءﺰﺟ رد يدﺎﻳز ﻲﺘﺨﺳ طﺎﻘﻧ د

ﺮﺗ ﻲﻘﻄﻨﻣ ﺞﻳﺎﺘﻧ ﺎﻨﺒﻣ ﺮﺘﻤﻛ تﺎﺒﺳﺎﺤﻣ ﻢﺠﺣ و

دﻮﺑ ﺪﻨﻫاﻮﺧ .

ﺖﻗﻮﻟا ﻦﺑا ﻞﻜﺷ ﺮﻴﻴﻐﺗ يﺎﻫدﻮﻣ ﺶﻳاﺪﻴﭘ ترﻮﺼﺑ تﻻﺎﺣ ﻦﻳا



)

hourglass modes

(

ﺖﺷاد ﺪﻫاﻮﺧ صﺎﺧ لﺮﺘﻨﻛ ﻪﺑ زﺎﻴﻧ ﻪﻛ ﺖﺳا .

ي

 The reduced-integration second-order serendipity interpolation elements in two dimensions—the 8-node quadrilaterals—have one such mode, but it is benign because it cannot propagate in a mesh with more than one element.

 The second-order three-dimensional elements with reduced integration have modes that can propagate in a single stack of elements. Because these modes rarely cause trouble in the second-order elements, no special techniques are used in ABAQUS to control them.

 In contrast, when reduced integration is used in the first-order elements (the 4-node quadrilateral and the 8-node brick), hourglassing can often make the elements unusable unless it is controlled.

 In ABAQUS the artificial stiffness method given in Flanagan and Belytschko (1981) is used to control the hourglass modes in these elements.

(18)

ي

The FE model will have no resistance to loads that activate these m The stiffness matrix will be singular. Dr. Hasan Ghasemzadeh 35

ي



Hourglass mode for 8-node element with

reduced integration to four points

(19)

ي

 The hourglass control methods of Flanagan and Belytschko (1981) are generally successful for linear and mildly nonlinear problems but may break down in strongly nonlinear problems and,

therefore, may not yield reasonable results.

 Success in controlling hourglassing also depends on the loads applied to the structure. For example, a point load is much more likely to trigger hourglassing than a distributed load.

 Hourglassing can be particularly troublesome in eigenvalue

extraction problems: the low stiffness of the hourglass modes may create many unrealistic modes with low eigenfrequencies.

 Experience suggests that the reduced-integration, second-order isoparametric elements are the most cost-effective elements in ABAQUS for problems in which the solution can be expected to be smooth.