The 12 reference contexts in paper A. Popov Yu., А. Попов Ю. (2016) “Алгоритмы численного анализа неклассических математических моделей термомеханики // Algorithms of the Numerical Analysis of Non-classical Thermomechanical Models” / spz:neicon:technomag:y:2015:i:6:p:83-98

  1. Start
    1916
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    ÎÓ͇Π̧ÌÓÒÚË ‚ ×— ÔÓÒÚÛÎËÛÂÚ, ̃ÚÓ Á̇ ̃ÂÌˡ ‡ÍÚË‚Ì ̊ı ÔÂÂÏÂÌÌ ̊ı (ÍÓÚÓ ̊ ı‡‡ÍÚÂËÁÛ ̨Ú ‚ÌÛÚÂÌÌ ÒÓÒÚÓˇÌË χÚÂˇ·; ̇ÔËÏÂ, ÚÂÌÁÓ ̇ÔˇÊÂÌËȈσ, ‚ÂÍÚÓ ÚÂÔÎÓ‚Ó„Ó ÔÓÚÓ͇q) ‚ ÓÍÂÒÚÌÓÒÚË ‡ÒÒχÚË‚‡ÂÏÓÈ ÚÓ ̃ÍË Á‡‚ËÒˇÚ ÚÓÎ ̧ÍÓ ÓÚ Á̇ ̃ÂÌËÈ ‡ÍÚË‚Ì ̊ı ÔÂÂÏÂÌÌ ̊ı (ÓÚ‡Ê‡ ̨Ú Â‡ÍˆË ̨ χÚÂˇ· ̇ ‚Ì ̄ÌË ‚ÓÁ‰ÂÈÒڂˡ; ̇ÔËÏÂ, ÚÂÏÔÂ‡ÚÛ‡, ̋ÎÂÍÚË ̃ÂÒÍËÈ Á‡ˇ‰)‚ ÓÍÂÒÚÌÓÒÚË ̋ÚÓÈ ÚÓ ̃ÍË
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    [1,2]
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    . ¬ ̃‡ÒÚÌÓÒÚË, Í·ÒÒË ̃ÂÒÍË ÔÓ‰ıÓ‰ ̊ ×— Á‡ ̃‡ÒÚÛ ̨ ÌÂÔËÏÂÌËÏ ̊ ÔË ÚÂÓÂÚË ̃ÂÒÍÓÏ ÔÓ„ÌÓÁËÓ‚‡ÌËË Ò‚ÓÈÒÚ‚ ÒÓ‚ÂÏÂÌÌ ̊ı ÍÓÌÒÚÛ͈ËÓÌÌ ̊ı Ë ÙÛÌ͈ËÓ̇Π̧Ì ̊ı χÚÂˇÎÓ‚. ƒÎˇ ÔÓËÁ‚Ó‰ÒÚ‚‡ Ú‡ÍËı χÚÂˇÎÓ‚ ËÒÔÓÎ ̧ÁÛ ̨ÚÒˇ ÒÔˆˇΠ̧Ì ̊ ÏÂÚÓ‰ ̊, ÔÓÁ‚ÓΡ ̨ ̆Ë ‰ÓÒÚË ̃ ̧ Ô‰ÂÎ ̧Ì ̊ı Á̇ ̃ÂÌËÈ Ëı ı‡‡ÍÚÂËÒÚËÍ: ‚ ̊ÒÓÍÓÈ ÔÓ ̃ÌÓÒÚË, ÚÂÔÎÓÂÏÍÓÒÚË, ÌËÁÍÓÈ ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË Ë Ô.
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  2. Start
    3915
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    ŒÌ Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ‡ÒÔÓÒÚ‡ÌÂÌËË ÏÂÚÓ‰Ó‚ ×—, ÓÔËÒ ̊‚‡ ̨ ̆Ëı Ôӂ‰ÂÌË χÚÂˇ· ̇ χÍÓÛÓ‚ÌÂ, ̇ ÏËÍÓÛÓ‚ÂÌ ̧ Ë ÔÓÒÎÂ‰Û ̨ ̆ÂÏ ÛÒÚ‡ÌÓ‚ÎÂÌËË Ò‚ˇÁË ÏÂÊ‰Û ı‡‡ÍÚÂËÒÚË͇ÏË Ï‡ÍÓ- Ë ÏËÍÓ- (ËÎË Ì‡ÌÓ-) ÛÓ‚Ìˇ. Õ‡ ÔËÏÂÌÂÌËË ÏÂÚÓ‰‡ ÌÂÔÂ ̊‚ÌÓÈ ‡ÔÔÓÍÒËχˆËË ÓÒÌÓ‚‡Ì‡ ӷ·ÒÚ ̧ ̇ÛÍË, ̇Á ̊‚‡ÂχˇÓ·Ó· ̆ÂÌÌÓÈ ÏÂı‡ÌËÍÓÈ ÒÔÎÓ ̄ÌÓÈ Ò‰ ̊
    Exact
    [3]
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    . ¬ÏÂÒÚÓ Í·ÒÒË ̃ÂÒÍËı (ÎÓ͇Π̧Ì ̊ı) χÚÂχÚË ̃ÂÒÍËı ÏÓ‰ÂÎÂÈ, ‚ ÌÂÈ ̄ËÓÍÓ ËÒÔÓÎ ̧ÁÛ ̨ÚÒˇ ÏÓ‰ÂÎË ÒÔÎÓ ̄ÌÓÈ Ò‰ ̊ Ò ÌÂÎÓ͇Π̧ÌÓÒÚ ̧ ̨, ‚ ̊ÒÚÛÔ‡ ̨ ̆Ë ӷӷ ̆ÂÌËÂÏ ÔÂ‚ ̊ı. ” ̃ËÚ ̊‚‡Âχˇ ÔË Ú‡ÍÓÏ ÔÓ‰ıӉ ÌÂÎÓ͇Π̧ÌÓÒÚ ̧ ̇Á ̊‚‡ÂÚÒˇÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ.
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  3. Start
    4883
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    ŒÒÌÓ‚Ì ̊ ÔÓÎÓÊÂÌˡ ÏÂı‡ÌËÍË ÒÔÎÓ ̄ÌÓÈ Ò‰ ̊ Ò ÌÂÎÓ͇Π̧ÌÓÒÚ ̧ ̨ œË ÔÓÒÚÓÂÌËË ÏÓ‰ÂÎÂÈ ÒÔÎÓ ̄ÌÓÈ Ò‰ ̊ Ò ÌÂÎÓ͇Π̧ÌÓÒÚ ̧ ̨ ÓÒÌÓ‚ ̊‚‡ ̨ÚÒˇ ̇ ÒÎÂ‰Û ̨ ̆Ëı ÔÓÎÓÊÂÌˡı, ˇ‚Ρ ̨ ̆ËıÒˇ Ó·Ó· ̆ÂÌËÂÏ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı ÎÓ͇Π̧Ì ̊ı ÔË̈ËÔÓ‚
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    [4]
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    : 1. «‡ÍÓÌ ÒÓı‡ÌÂÌˡ ̋ÌÂ„ËË (1) ‚ ̊ÔÓÎÌˇÂÚÒˇ ÚÓÎ ̧ÍÓ ‚ ËÌÚ„‡Î ̧ÌÓÈ ÙÓÏ (‰Îˇ ‚ÒÂ„Ó Ú·): d dt ∫ V ρ ( ε+ 1 2 vivi ) dV= ∫ S (σklvl+qk)nkdS+ ∫ V ρ(fkvk+h)dV,(1) „‰Âε| χÒÒÓ‚‡ˇ ÔÎÓÚÌÓÒÚ ̧ ‚ÌÛÚÂÌÌÂÈ ̋ÌÂ„ËË Ú·;vi| ÍÓÏÔÓÌÂÌÚ ̊ ÔÓΡ ÒÍÓÓÒÚÂÈ;fk| ÍÓÏÔÓÌÂÌÚ ̊ ‚ÂÍÚÓ‡ χÒÒÓ‚ÓÈ ÔÎÓÚÌÓÒÚË ‚Ì ̄ÌËı ÒËÎ;h| χÒÒÓ‚‡ˇ ÔÎÓÚÌÓÒÚ ̧ ËÒÚÓ ̃ÌËÍÓ‚ ÚÂÔÎÓÚ ̊. 2.
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  4. Start
    5570
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    Á̇ ̃ÂÌˡ ‡ÍÚË‚Ì ̊ı ÔÂÂÏÂÌÌ ̊ı) ÔÓËÁ‚ÓÎ ̧ÌÓÈ ÚÓ ̃ÍË Ú· ‚ ‰‡ÌÌ ̊È ÏÓÏÂÌÚ Á‡‚ËÒËÚ ÓÚ ÒÓÒÚÓˇÌˡ ‚ÒÂı ÓÒڇΠ̧Ì ̊ı ÚÓ ̃ÂÍ Ú· ‚ ÚÂÍÛ ̆ËÈ ÏÓÏÂÌÚ, ‡ Ú‡ÍÊ ‚Ó ‚Ò Ô‰ ̊‰Û ̆ËÂ. ¬ ‰ÓÔÓÎÌÂÌËÂ Í ÓÒÌÓ‚Ì ̊Ï ÓÔ‰ÂΡ ̨ ̆ËÏ ÒÓÓÚÌÓ ̄ÂÌËˇÏ Ë ÔÓÎÓÊÂÌËˇÏ ÏÂı‡ÌËÍË ÒÔÎÓ ̄ÌÓÈ Ò‰ ̊ Ò ÌÂÎÓ͇Π̧ÌÓÒÚ ̧ ̨ Ô‰ÎÓÊÂÌ
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    [4]
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    ̇·Ó ÔË̈ËÔÓ‚, ËÒÔÓÎ ̧ÁÛÂÏ ̊ı ‰Îˇ ‡ÒÔÓÒÚ‡ÌÂÌˡ ÏÂÚÓ‰ÓÎÓ„ËË ÍÓÌÚËÌÛÛχ ̇ ÏËÍÓÛÓ‚ÂÌ ̧. —Â‰Ë ̋ÚËı ÔË̈ËÔÓ‚ ÓÚÏÂÚËÏ ÔË̈ËÔ ÒÓÒ‰ÒÚ‚‡ Ë ÔË̈ËÔ Ô‡ÏˇÚË. —ӄ·ÒÌÓ ÔÂ‚ÓÏÛ, ̃ÂÏ ‰‡Î ̧ ̄ ÌÂÍÓÚÓ‡ˇ ÚÓ ͇̃r′ÓÚ ‡ÒÒχÚË‚‡ÂÏÓÈr, ÚÂÏ Ò··Â ‚ÎˡÌË ÒÓÒÚÓˇÌˡ Ú· ‚r′̇ Â„Ó ÒÓÒÚÓˇÌË ‚r.
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  5. Start
    6744
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    ÇÚÂχÚË ̃ÂÒ͇ˇ ÏÓ‰ÂÎ ̧ ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË, Û ̃ËÚ ̊‚‡ ̨ ̆‡ˇ ÌÂÎÓ͇Π̧ÌÓÒÚ ̧ ÔÓ ‚ÂÏÂÌË ŒÒÌÓ‚Ì ̊ ÒÓÓÚÌÓ ̄ÂÌˡ.–‡ÒÔ‰ÂÎÂÌË ÚÂÏÔÂ‡ÚÛ ̊ ‚ Ú‚Â‰ÓÏ ÚÂΠ‚ ÏÓ‰ÂÎË ÒÔÎÓ ̄ÌÓÈ Ò‰ ̊ Ò ‚ÌÛÚÂÌÌËÏË Ô‡‡ÏÂÚ‡ÏË ÒÓÒÚÓˇÌˡ ÏÓÊÂÚ · ̊Ú ̧ ÓÔËÒ‡ÌÓ ÒÎÂ‰Û ̨ ̆ËÏ Û‡‚ÌÂÌËÂÏ
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    [5,6]
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    : ρc ∂T ∂t +ρc ( ∂T ∂t − ∫t 0 ∂2T ∂t′2 exp ( − t−t′ τt ) dt′ ) = = ∂ ∂xi ( λ (T) ij ( ∂T ∂xj − ∫t 0 ∂ ∂t′ ∂T ∂xj exp ( − t−t′ τq ) dt′ )) ,(2) „‰Âλ (T) ij| ÍÓÏÔÓÌÂÌÚ ̊ ÚÂÌÁÓ‡ ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË;c=cε| Û‰ÂÎ ̧̇ˇ χÒÒÓ‚‡ˇ ÚÂÔÎÓÂÏÍÓÒÚ ̧ ÔË ÔÓÒÚÓˇÌÌÓÈ ‰ÂÙÓχˆËË,τtËτq| ‚ÂÏÂ̇ ·ÍÒ‡ˆËË.
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  6. Start
    7945
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    4) Û‡‚ÌÂÌË (2) ÔËÏÂÚ ‚ˉ ρc ∂T ∂t +ρc ∂T ∂t ∣ ∣ ∣ ∣ ∣ t=0 exp ( − t τt ) + ρc τt ∫t 0 ∂T ∂t′ exp ( − t−t′ τt ) dt′= = ∂ ∂xi ( λ (T) ij ∂T ∂xj ∣ ∣ ∣ ∣∣ t=0 exp ( − t τq ) + λ (T) ij τq ∫t 0 ∂T ∂xj exp ( − t−t′ τq ) dt′ ) .(5) ›ÚÓ Û‡‚ÌÂÌËÂ, ‚ ÓÚÎË ̃Ë ÓÚ Í·ÒÒË ̃ÂÒÍÓ„Ó Ô‡‡·ÓÎË ̃ÂÒÍÓ„Ó Û‡‚ÌÂÌˡ ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË ρc ∂T ∂t = ∂ ∂xi ( λ (T) ij ∂T ∂xj ) , ˇ‚ΡÂÚÒˇ Û‡‚ÌÂÌËÂÏ „ËÔÂ·ÓÎË ̃ÂÒÍÓ„Ó ÚËÔ‡
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    [8]
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    . ”‡‚ÌÂÌË (5) ÒÓ‰ÂÊËÚ ‰ÓÔÓÎÌËÚÂÎ ̧Ì ̊ Ò·„‡ÂÏ ̊Â, Û ̃ËÚ ̊‚‡ ̨ ̆Ë ÌÂÎÓ͇Π̧ÌÓÒÚ ̧ ÔÓ ‚ÂÏÂÌË: Ò·„‡ÂÏ ̊ ‚ ΂ÓÈ ̃‡ÒÚË ‡‚ÂÌÒÚ‚‡ ÓÚ‡Ê‡ ̨Ú ̋ÙÙÂÍÚ Á‡Ô‡Á‰ ̊‚‡Ìˡ ÔË ‡ÍÍÛÏÛΡˆËË ÚÂÔÎÓÚ ̊, ‚ Ô‡‚ÓÈ | ÍÓÌ ̃ÌÛ ̨ ÒÍÓÓÒÚ ̧ ‡ÒÔÓÒÚ‡ÌÂÌˡ ÚÂÔÎÓÚ ̊ ‚ ÚÂÎÂ. «‡ ̃‡ÒÚÛ ̨ χÚÂχÚË ̃ÂÒÍË ÏÓ‰ÂÎË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË Ò ÌÂÎÓ͇Π̧ÌÓÒÚ ̧ ̨ ËÒÔÓÎ ̧ÁÛ ̨ÚÒˇ ‰Îˇ ÏÓ‰ÂÎËÓ‚‡Ìˡ ÔÓˆÂÒÒÓ‚ ‚ ̊ÒÓÍÓËÌÚÂÌÒË‚ÌÓ„Ó Ì‡„‚‡ [
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  7. Start
    8342
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    ”‡‚ÌÂÌË (5) ÒÓ‰ÂÊËÚ ‰ÓÔÓÎÌËÚÂÎ ̧Ì ̊ Ò·„‡ÂÏ ̊Â, Û ̃ËÚ ̊‚‡ ̨ ̆Ë ÌÂÎÓ͇Π̧ÌÓÒÚ ̧ ÔÓ ‚ÂÏÂÌË: Ò·„‡ÂÏ ̊ ‚ ΂ÓÈ ̃‡ÒÚË ‡‚ÂÌÒÚ‚‡ ÓÚ‡Ê‡ ̨Ú ̋ÙÙÂÍÚ Á‡Ô‡Á‰ ̊‚‡Ìˡ ÔË ‡ÍÍÛÏÛΡˆËË ÚÂÔÎÓÚ ̊, ‚ Ô‡‚ÓÈ | ÍÓÌ ̃ÌÛ ̨ ÒÍÓÓÒÚ ̧ ‡ÒÔÓÒÚ‡ÌÂÌˡ ÚÂÔÎÓÚ ̊ ‚ ÚÂÎÂ. «‡ ̃‡ÒÚÛ ̨ χÚÂχÚË ̃ÂÒÍË ÏÓ‰ÂÎË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË Ò ÌÂÎÓ͇Π̧ÌÓÒÚ ̧ ̨ ËÒÔÓÎ ̧ÁÛ ̨ÚÒˇ ‰Îˇ ÏÓ‰ÂÎËÓ‚‡Ìˡ ÔÓˆÂÒÒÓ‚ ‚ ̊ÒÓÍÓËÌÚÂÌÒË‚ÌÓ„Ó Ì‡„‚‡
    Exact
    [9,10,11,12,13,14]
    Suffix
    . “Ó„‰‡ Û‡‚ÌÂÌË (5) ÏÓÊÂÚ ‰ÓÔÓÎÌˇÚ ̧Òˇ „‡ÌË ̃Ì ̊Ï ÛÒÎÓ‚ËÂÏ ‚ˉ‡ (‚ Ó‰ÌÓÏÂÌÓÏ ÒÎÛ ̃‡Â) −λ(T) ( ∂T(x1, t) ∂x1 − ∫t 0 ∂ ∂t′ ∂T(x1, t′) ∂x1 exp ( − t−t′ τq ) dt′ )∣ ∣ ∣ ∣ ∣ x1=x(0)1 =qs, „‰Âqs| Á‡‰‡ÌÌ ̊È ÚÂÔÎÓ‚ÓÈ ÔÓÚÓÍ Ì‡ „‡ÌˈÂ. ¬ ̃‡ÒÚÌÓÒÚË, ÔËqs=BM tmexp (−mt/t0),B, M= const,m>1,m∈N, ÒÛ ̆ÂÒÚ‚ÛÂÚ ‡Ì‡ÎËÚË ̃ÂÒÍÓ  ̄ÂÌË Û‡‚ÌÂÌˡ ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË (5) [6]. 4.
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  8. Start
    8689
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    “Ó„‰‡ Û‡‚ÌÂÌË (5) ÏÓÊÂÚ ‰ÓÔÓÎÌˇÚ ̧Òˇ „‡ÌË ̃Ì ̊Ï ÛÒÎÓ‚ËÂÏ ‚ˉ‡ (‚ Ó‰ÌÓÏÂÌÓÏ ÒÎÛ ̃‡Â) −λ(T) ( ∂T(x1, t) ∂x1 − ∫t 0 ∂ ∂t′ ∂T(x1, t′) ∂x1 exp ( − t−t′ τq ) dt′ )∣ ∣ ∣ ∣ ∣ x1=x(0)1 =qs, „‰Âqs| Á‡‰‡ÌÌ ̊È ÚÂÔÎÓ‚ÓÈ ÔÓÚÓÍ Ì‡ „‡ÌˈÂ. ¬ ̃‡ÒÚÌÓÒÚË, ÔËqs=BM tmexp (−mt/t0),B, M= const,m>1,m∈N, ÒÛ ̆ÂÒÚ‚ÛÂÚ ‡Ì‡ÎËÚË ̃ÂÒÍÓ  ̄ÂÌË Û‡‚ÌÂÌˡ ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË (5)
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    [6]
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    . 4. ÇÚÂχÚË ̃ÂÒ͇ˇ ÏÓ‰ÂÎ ̧, Û ̃ËÚ ̊‚‡ ̨ ̆‡ˇ ̋ÙÙÂÍÚ Á‡Ô‡Á‰ ̊‚‡Ìˡ ÔË ‡ÍÍÛÏÛΡˆËË ÚÂÔÎÓÚ ̊ ”‡‚ÌÂÌË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË.ƒÎˇ ̃ËÒÎÂÌÌÓ„Ó ÏÓ‰ÂÎËÓ‚‡Ìˡ ÒÔÎÓ ̄ÌÓÈ Ò‰ ̊ Ò Ô‡ÏˇÚ ̧ ̨ ˆÂÎÂÒÓÓ·‡ÁÌÓ ÓÚ‰ÂÎ ̧ÌÓ ‡ÒÒÏÓÚÂÚ ̧ χÚÂχÚË ̃ÂÒÍË ÏÓ‰ÂÎË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË, Û ̃ËÚ ̊‚‡ ̨ ̆Ë ‰‚‡ ‡ÁÎË ̃Ì ̊ı ̋ÙÙÂÍÚ‡.
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  9. Start
    9468
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    Á‰ ̊‚‡Ìˡ ÔË ‡ÍÍÛÏÛΡˆËË ÚÂÔÎÓÚ ̊, ÔË ̋ÚÓÏ ÒÍÓÓÒÚ ̧ ‡ÒÔÓÒÚ‡ÌÂÌˡ ÚÂÔÎÓÚ ̊ Ò ̃ËÚ‡ÂÚÒˇ ·ÂÒÍÓÌ ̃ÌÓÈ. ƒÎˇ Ó‰ÌÓÓ‰ÌÓ„Ó ËÁÓÚÓÔÌÓ„Ó Ú‚Â‰Ó„Ó Ú·λ (T) ij=λ (T)δ ij,a 2=λ (T) ρc | ÍÓ ̋ÙÙˈËÂÌÚ ÚÂÏÔÂ‡ÚÛÓÔÓ‚Ó‰ÌÓÒÚË Ú·. ÓÏ ÚÓ„Ó, ÔË ÏÓ‰ÂÎËÓ‚‡ÌËË ÔÓ‚ÂıÌÓÒÚÌÓ„Ó Ì‡„‚‡ ̃‡ÒÚÓ ËÏÂÂÚ ÏÂÒÚÓ Ì‡ ̃‡Î ̧ÌÓ ÛÒÎÓ‚Ë ∂T(x,t) ∂t ∣ ∣ ∣ ∣ t=0 = 0. —ÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ Û‡‚ÌÂÌË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË ËÏÂÂÚ ‚ˉ
    Exact
    [7]
    Suffix
    ∂T ∂t + 1 τt ∫t 0 ∂T ∂t′ exp ( − t−t′ τt ) dt′=a2 ∂2T ∂xj∂xj .(6) ≈ÒÎË ‚ÁˇÚ ̧ ÔÓ ̃‡ÒÚˇÏ ËÌÚ„‡Î ‚ ΂ÓÈ ̃‡ÒÚË Û‡‚ÌÂÌˡ (6), ÔÓÎÛ ̃ËÏ ∂T ∂t + 1 τt ( T(t)−T0(x) exp ( − t τt )) − 1 τ2t ∫t 0 T(t′) exp ( − t−t′ τt ) dt′=a2 ∂2T ∂xj∂xj ,(7) „‰ÂT0(x) =T(x, t) ∣ ∣ ∣ t=0 | ̇ ̃‡Î ̧ÌÓ ‡ÒÔ‰ÂÎÂÌË ÚÂÏÔÂ‡ÚÛ ̊.
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  10. Start
    10426
    Prefix
    – ̄ÂÌË Á‡‰‡ ̃Ë ‚ OpenFOAM ÔÓËÒıÓ‰ËÚ ‚ Ó· ̆ÂÏ ÒÎÛ ̃‡Â ‚ ÚÂıÏÂÌÓÈ ÔÓÒÚ‡ÌÓ‚ÍÂ, ÔË ̋ÚÓÏ ‰Îˇ  ̄ÂÌˡ Ó‰ÌÓ- ËÎË ‰‚ÛÏÂÌ ̊ı Á‡‰‡ ̃ ÌÂÓ·ıÓ‰ËÏÓ Û͇Á‡Ú ̧, ‚ ͇ÍËı ̇Ô‡‚ÎÂÌˡı ËÒÍÓχˇ ‚ÂÎË ̃Ë̇ Ì ËÁÏÂÌˇÂÚÒˇ. —ÚÓËÚ ÓÚÏÂÚËÚ ̧, ̃ÚÓ  ̄ÂÌË ̋ÚÓÈ Á‡‰‡ ̃Ë Ë ÔÓ‰Ó·Ì ̊ı ÂÈ (̇ÔËÏÂ,
    Exact
    [15]
    Suffix
    ) ‚ÓÁÏÓÊÌÓ Ë Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ ÏÂÚÓ‰‡ ÍÓÌ ̃Ì ̊ı ̋ÎÂÏÂÌÚÓ‚, Ӊ̇ÍÓ ‚ Ú‡ÍÓÏ ÒÎÛ ̃‡Â Ì ·Û‰ÂÚ Ó·ÂÒÔ ̃Ë‚‡Ú ̧Òˇ ‚ ̊ÔÓÎÌÂÌË ‰ËÒÍÂÚÌÓ„Ó ‡Ì‡ÎÓ„‡ Á‡ÍÓ̇ ÒÓı‡ÌÂÌˡ ̋ÌÂ„ËË. œË ‡ÔÔÓÍÒËχˆËË Û‡‚ÌÂÌˡ (7) ̇˷ÓÎ ̧ ̄ËÈ ËÌÚÂÂÒ Ô‰ÒÚ‡‚ΡÂÚ ËÌÚ„‡Î ̧ÌÓ Ò·„‡ÂÏÓ ‚ ΂ÓÈ ̃‡ÒÚË, ÍÓÚÓÓÂ, ÒÓ·ÒÚ‚ÂÌÌÓ, Û ̃ËÚ ̊‚‡ÂÚ ‚ÎˡÌË ‡ÒÔ‰ÂÎÂÌˡ ÚÂÏÔÂ‡ÚÛ ̊ ‚ ÚÂΠ‚ Ô‰ ̊‰Û ̆Ë ÏÓÏÂÌÚ ̊ ‚ÂÏÂÌË Ì‡ ÚÂÍÛ ̆ ‚ ÒÓ„
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  11. Start
    12769
    Prefix
    ¿ÔÔÓÍÒËχˆËˇ Ò·„‡ÂÏ ̊ı Û‡‚ÌÂÌˡ (7) ‚ OpenFOAM ÓÒÛ ̆ÂÒڂΡÂÚÒˇ ÓÔËÒ‡ÌÌ ̊Ï ÌËÊ ÒÔÓÒÓ·ÓÏ. ŒÔËÒ‡ÌÌ ̊ ‰‡Î ÔÓ‰ıÓ‰ ̊ Ì ˇ‚Ρ ̨ÚÒˇ ‰ËÌÒÚ‚ÂÌÌÓ ‚ÓÁÏÓÊÌ ̊ÏË ‚ OpenFOAM Ë ‰Îˇ ÔÓÒÚÓÂÌˡ —À¿” ÏÓ„ÛÚ · ̊Ú ̧ ÔËÏÂÌÂÌ ̊ ËÌ ̊ ÒıÂÏ ̊ ‡ÔÔÓÍÒËχˆËË Û‡‚ÌÂÌËÈ
    Exact
    [16]
    Suffix
    . 1. —·„‡ÂÏÓ ∂T ∂t : ∫ V0 ∫t t−∆t ∂T ∂t dt dV≈ ∫ V0 T−ˇT ∆t ∆t dV= (T−ˇT)V0, „‰ÂT=T(t)| ÌÂËÁ‚ÂÒÚ̇ˇ ÚÂÏÔÂ‡ÚÛ‡;ˇT=T(t−∆t). «Ì‡ ̃ÂÌˡ ÚÂÏÔÂ‡ÚÛ ̊ (T,ˇTË Ú.‰.) ·ÂÛÚÒˇ ‚ „ÂÓÏÂÚË ̃ÂÒÍËı ˆÂÌÚ‡ı ÍÓÌÚÓÎ ̧Ì ̊ı Ó· ̇ÂÏÓ‚. 2.
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  12. Start
    16568
    Prefix
    ÇÚÂχÚË ̃ÂÒ͇ˇ ÏÓ‰ÂÎ ̧ ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË, Û ̃ËÚ ̊‚‡ ̨ ̆‡ˇ ÌÂÎÓ͇Π̧ÌÓÒÚ ̧ ÔÓ ÔÓÒÚ‡ÌÒÚ‚Û ”‡‚ÌÂÌË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË.–‡ÒÔ‰ÂÎÂÌË ÚÂÏÔÂ‡ÚÛ ̊ ‚ χÚÂˇÎÂ Ò ÌÂÎÓ͇Π̧ÌÓÒÚ ̧ ̨ ÔÓ ÔÓÒÚ‡ÌÒÚ‚Û ÓÔËÒ ̊‚‡ÂÚÒˇ ÒÎÂ‰Û ̨ ̆ËÏ Û‡‚ÌÂÌËÂÏ
    Exact
    [6,17]
    Suffix
    : ρc A44 ( γ1 ∂T(x, t) ∂t +γ2 ∫ V φT(|x′−x|) ∂T(x′, t) ∂t dV(x′) ) = = ∂ ∂xi ( λ (T) ij ( γ1 ∂T ∂xj +γ2 ∫ V φ1(|x′−x|) ∫ V φ2(|x′′−x′|) ∂T ∂xj dV(x′′)dV(x′) )) .(11) «‰ÂÒ ̧γ1, γ2∈[0; 1],γ1+γ2= 1, | ÍÓ ̋ÙÙˈËÂÌÚ ̊ Û ̃ÂÚ‡ ‡ÁÎË ̃Ì ̊ı ÔÓˆÂÒÒÓ‚ ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË ‚ ÚÂÎÂ, ÔË ̃ÂÏγ1| ‰ÓΡ ÎÓ͇Π̧ÌÓ„Ó ÚÂÔÎÓÓ·ÏÂ̇ (ÏÂÊ‰Û ÒÓÒ‰ÌËÏË ÒÚÛÍÚÛ‡ÏË Ï‡ÚÂˇ·),γ2| ÍÓ ̋ÙÙˈËÂÌÚ Û ̃ÂÚ‡ ‚ÎˡÌˡ ̇ ÓÔ‰ÂÎÂÌÌÛ ̨ ÒÚÛ
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