RD-E: 4300 Perfect Gas Modeling with Polynomial EOS

The purpose of this example is to plot numerical pressure, internal energy, and sound speed for a perfect gas material law.

Comparison to theoretical results is made. Control cards for Absolute and Relative formulations will be used.

ex43_perfect_gas_model
Figure 1.
Polynomial EOS is often used by Radioss to compute hydrodynamic pressure and used to model perfect gas. It is cubic in compression and linear in expansion.(1) P= C 0 + C 1 μ+ C 2 μ 2 + C 3 μ 3 +( C 4 + C 5 μ )E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0Jaam4qamaaBaaaleaacaaIWaaabeaakiabgUcaRiaadoeadaWg aaWcbaGaaGymaaqabaGccqaH8oqBcqGHRaWkcaWGdbWaaSbaaSqaai aaikdaaeqaaOGaeqiVd02aaWbaaSqabeaacaaIYaaaaOGaey4kaSIa am4qamaaBaaaleaacaaIZaaabeaakiabeY7aTnaaCaaaleqabaGaaG 4maaaakiabgUcaRmaabmaabaGaam4qamaaBaaaleaacaaI0aaabeaa kiabgUcaRiaadoeadaWgaaWcbaGaaGynaaqabaGccqaH8oqBaiaawI cacaGLPaaacaWGfbaaaa@531D@
Where,(2) E = E int V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGfbGaeyypa0ZaaSaaaeaacaWGfbWaaSbaaSqaaiGacMgacaGG UbGaaiiDaaqabaaakeaacaWGwbWaaSbaaSqaaiaaicdaaeqaaaaaaa a@4051@
and (3) μ = ρ ρ 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaH8oqBcqGH9aqpdaWcaaqaaiabeg8aYbqaaiabeg8aYnaaBaaa leaacaaIWaaabeaaaaGccqGHsislcaaIXaaaaa@41BB@

A simple test of compression/expansion is made to compare these formulation outputs with theoretical results.

Options and Keywords Used

Input Files

Refer to Access the Model Files to download the required model file(s).

Model Description

This test consists with an elementary volume of perfect gas undergoing spherical expansion and compression.

ex43_cube
Figure 2.
Initial conditions are:
  • P 0 = 1 e 5 [ Pa ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS baaSqaaiaaicdaaeqaaOGaeyypa0JaaGymaiaadwgacaaI1aWaamWa aeaaciGGqbGaaiyyaaGaay5waiaaw2faaaaa@4043@
  • V 0 = 1000 [ m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGwbWaaS baaSqaaiaaicdaaeqaaOGaeyypa0JaaGymaiaaicdacaaIWaGaaGim amaadmaabaGaaiyBamaaCaaaleqabaGaai4maaaaaOGaay5waiaaw2 faaaaa@40F2@
  • ρ 0 = 1.204 [ k g m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCda WgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIXaGaaiOlaiaaikdacaaI WaGaaGinamaadmaabaWaaSaaaeaacaGGRbGaai4zaaqaaiaac2gada ahaaWcbeqaaiaacodaaaaaaaGccaGLBbGaayzxaaaaaa@4479@
  • μ 0 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda WgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIWaaaaa@3BCD@

The fluid will be assumed to be a perfect gas. Volume is changed in the three directions to consider a pure compression 1 < μ < 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislca aIXaGaeyipaWJaeqiVd0MaeyipaWJaaGimaaaa@3D87@ followed by an expansion of matter μ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBcq GH+aGpcaaIWaaaaa@3ADF@ (Figure 3).

This test will be modeled with a single ALE element (8 node brick) and polynomial EOS.

Evolutions of pressure, internal energy and sound speed will be compared between numerical output and theoretical results.

The length is modified with /IMPDISP; its influences on V and μ are plotted (Figure 3).

ex43_elementary_volume_change
Figure 3. Elementary volume change

Simulation Iterations

A single ALE brick element is used. Material is confined inside the element by defining brick nodes as Lagrangian. For each face, displacement is imposed on the four nodes along the normal.

Polynomial EOS

Polynomial EOS is used in /EOS/POLYNOMIAL to compute hydrodynamic pressure. It is cubic in compression and linear in expansion.(4) P= C 0 + C 1 μ+ C 2 μ 2 + C 3 μ 3 +( C 4 + C 5 μ )E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0Jaam4qamaaBaaaleaacaaIWaaabeaakiabgUcaRiaadoeadaWg aaWcbaGaaGymaaqabaGccqaH8oqBcqGHRaWkcaWGdbWaaSbaaSqaai aaikdaaeqaaOGaeqiVd02aaWbaaSqabeaacaaIYaaaaOGaey4kaSIa am4qamaaBaaaleaacaaIZaaabeaakiabeY7aTnaaCaaaleqabaGaaG 4maaaakiabgUcaRmaabmaabaGaam4qamaaBaaaleaacaaI0aaabeaa kiabgUcaRiaadoeadaWgaaWcbaGaaGynaaqabaGccqaH8oqBaiaawI cacaGLPaaacaWGfbaaaa@531D@
C parameters are called hydrodynamic coefficients and they are input parameters. Hypothesis on the material behavior allows determining of these coefficients:
  • Incompressible gas
  • Linear elastic material
  • Perfect gas

This example is focused only on Perfect Gas modeling.

Results

Theoretical Results

The purpose of this section is to plot pressure, internal energy, and sound speed in function of the single parameter V or μ .
  1. Pressure:
    Perfect gas pressure is given by:(5) P V = ( γ 1 ) E int MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGqbGaamOvaiabg2da9maabmaabaGaeq4SdCMaeyOeI0IaaGym aaGaayjkaiaawMcaaiaadweadaWgaaWcbaGaciyAaiaac6gacaGG0b aabeaaaaa@4434@
    Then,(6) d P ( V , E int ) = P V | E int d V + P E int | V d E int MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam iuamaabmaabaGaamOvaiaacYcacaWGfbWaaSbaaSqaaiGacMgacaGG UbGaaiiDaaqabaaakiaawIcacaGLPaaacqGH9aqpdaabcaqaamaala aabaGaeyOaIyRaamiuaaqaaiabgkGi2kaadAfaaaaacaGLiWoadaWg aaWcbaGaamyramaaBaaameaaciGGPbGaaiOBaiaacshaaeqaaaWcbe aakiaadsgacaWGwbGaey4kaSYaaqGaaeaadaWcaaqaaiabgkGi2kaa dcfaaeaacqGHciITcaWGfbWaaSbaaSqaaiGacMgacaGGUbGaaiiDaa qabaaaaaGccaGLiWoadaWgaaWcbaGaamOvaaqabaGccaWGKbGaamyr amaaBaaaleaaciGGPbGaaiOBaiaacshaaeqaaaaa@5CDA@
    Radioss assumes the hypothesis of an isentropic process to compute the change in internal energy:(7) d E int = P d V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam yramaaBaaaleaaciGGPbGaaiOBaiaacshaaeqaaOGaeyypa0JaeyOe I0IaamiuaiaadsgacaWGwbaaaa@40B5@
    This theory gives the following differential equation:(8) d P d V = γ P V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai aadsgacaWGqbaabaGaamizaiaadAfaaaGaeyypa0JaeyOeI0YaaSaa aeaacqaHZoWzcaWGqbaabaGaamOvaaaaaaa@4053@
    This has the form y ' + γ x = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bGaai 4jaiabgUcaRmaalaaabaGaeq4SdCgabaGaamiEaaaacqGH9aqpcaaI Waaaaa@3E66@ and the general solution is:(9) y = C s t . x γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWG5bGaeyypa0Jaam4qaiaadohacaWG0bGaaiOlaiaadIhadaah aaWcbeqaaiabgkHiTiabeo7aNbaaaaa@4204@
    Pressure is also polytropic:(10) P V γ = P 0 V 0 γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaam OvamaaCaaaleqabaGaeq4SdCgaaOGaeyypa0JaamiuamaaBaaaleaa caaIWaaabeaakiaadAfadaWgaaWcbaGaaGimaaqabaGcdaahaaWcbe qaaiabeo7aNbaaaaa@415F@ (11) P ( V ) = P 0 ( V 0 V ) γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGqbWaaeWaaeaacaWGwbaacaGLOaGaayzkaaGaeyypa0Jaamiu amaaBaaaleaacaaIWaaabeaakmaabmaabaWaaSaaaeaacaWGwbWaaS baaSqaaiaaicdaaeqaaaGcbaGaamOvaaaaaiaawIcacaGLPaaadaah aaWcbeqaaiabeo7aNbaaaaa@44EE@

    Here, γ is the material constant (ratio of heat capacity). For diatomic gas γ =1.4. Air is made mainly of diatomic gas, so set gamma to 1.4 for air.

  2. Internal Energy:
    Equation 5 and Equation 11 lead to the immediate result:(12) E int ( V ) = P 0 V 0 γ ( γ 1 ) V γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGfbWaaSbaaSqaaiGacMgacaGGUbGaaiiDaaqabaGcdaqadaqa aiaadAfaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaadcfadaWgaa WcbaGaaGimaaqabaGccaWGwbWaaSbaaSqaaiaaicdaaeqaaOWaaWba aSqabeaacqaHZoWzaaaakeaadaqadaqaaiabeo7aNjabgkHiTiaaig daaiaawIcacaGLPaaacaWGwbWaaWbaaSqabeaacqaHZoWzcqGHsisl caaIXaaaaaaaaaa@4EC7@
  3. Sound Speed:
    Perfect gas sound speed is:(13) c = γ P ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbGaey ypa0ZaaOaaaeaadaWcaaqaaiabeo7aNjaadcfaaeaacqaHbpGCaaaa leqaaaaa@3DBC@
    Equation 11 gives its expression in term of volume:(14) c = γ P 0 ρ 0 ( V 0 V ) γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGJbGaeyypa0ZaaSaaaeaacqaHZoWzcaWGqbWaaSraaSqaaiaa icdaaeqaaaGcbaGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaaaakmaabm aabaWaaSaaaeaacaWGwbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaamOv aaaaaiaawIcacaGLPaaadaahaaWcbeqaaiabeo7aNjabgkHiTiaaig daaaaaaa@48AD@
Pressure, internal energy, and sound speed are expressed both in function of V and μ .
Table 1. Theoretical Results
Pressure (Pa) Internal Energy Density (J) Sound Speed (m/s)
P R E F ( V ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaW baaSqabeaacaWGsbGaamyraiaadAeaaaGcdaqadaqaaiaadAfaaiaa wIcacaGLPaaaaaa@3D44@ P R E F ( μ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaW baaSqabeaacaWGsbGaamyraiaadAeaaaGcdaqadaqaaiabeY7aTbGa ayjkaiaawMcaaaaa@3E1F@ ρ e R E F ( V ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCca qGLbWaaWbaaSqabeaacaWGsbGaamyraiaadAeaaaGcdaqadaqaaiaa dAfaaiaawIcacaGLPaaaaaa@3F17@ ρ e R E F ( μ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCca qGLbWaaWbaaSqabeaacaWGsbGaamyraiaadAeaaaGcdaqadaqaaiab eY7aTbGaayjkaiaawMcaaaaa@3FF2@ c R E F ( V ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaW baaSqabeaacaWGsbGaamyraiaadAeaaaGcdaqadaqaaiaadAfaaiaa wIcacaGLPaaaaaa@3D57@ c R E F ( μ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaW baaSqabeaacaWGsbGaamyraiaadAeaaaGcdaqadaqaaiabeY7aTbGa ayjkaiaawMcaaaaa@3E32@
P 0 ( V 0 V ) γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGqbWaaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaadaWcaaqaaiaa dAfadaWgaaWcbaGaaGimaaqabaaakeaacaWGwbaaaaGaayjkaiaawM caamaaCaaaleqabaGaeq4SdCgaaaaa@40AF@ P 0 ( 1 + μ ) γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGqbWaaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacaaIXaGaey4k aSIaeqiVd0gacaGLOaGaayzkaaWaaWbaaSqabeaacqaHZoWzaaaaaa@414C@ P 0 ( γ 1 ) ( V 0 V ) γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaWcaaqaaiaadcfadaWgaaWcbaGaaGimaaqabaaakeaadaqadaqa aiabeo7aNjabgkHiTiaaigdaaiaawIcacaGLPaaaaaWaaeWaaeaada WcaaqaaiaadAfadaWgaaWcbaGaaGimaaqabaaakeaacaWGwbaaaaGa ayjkaiaawMcaamaaCaaaleqabaGaeq4SdCMaeyOeI0IaaGymaaaaaa a@473F@ P 0 ( γ 1 ) ( 1 + μ ) γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaWcaaqaaiaadcfadaWgaaWcbaGaaGimaaqabaaakeaadaqadaqa aiabeo7aNjabgkHiTiaaigdaaiaawIcacaGLPaaaaaWaaeWaaeaaca aIXaGaey4kaSIaeqiVd0gacaGLOaGaayzkaaWaaWbaaSqabeaacqaH ZoWzcqGHsislcaaIXaaaaaaa@47DC@ γ P 0 ρ 0 ( V 0 V ) γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaGcaaqaamaalaaabaGaeq4SdCMaamiuamaaBaaaleaacaaIWaaa beaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaaaGcdaqadaqaam aalaaabaGaamOvamaaBaaaleaacaaIWaaabeaaaOqaaiaadAfaaaaa caGLOaGaayzkaaWaaWbaaSqabeaacqaHZoWzcqGHsislcaaIXaaaaa qabaaaaa@46CE@ γ P 0 ρ 0 ( 1 + μ ) γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaGcaaqaamaalaaabaGaeq4SdCMaamiuamaaBaaaleaacaaIWaaa beaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaaaGcdaqadaqaai aaigdacqGHRaWkcqaH8oqBaiaawIcacaGLPaaadaahaaWcbeqaaiab eo7aNjabgkHiTiaaigdaaaaabeaaaaa@476B@
Corresponding plots are:

ex43_perfect_gas_pressure
Figure 4. Perfect Gas Pressure

ex43_perfect_gas_internal_energy
Figure 5. Perfect Gas Internal Energy

ex43_perfect_gas_sound_speed
Figure 6. Perfect Gas Sound Speed

Material Control Cards

/MAT/LAW6 (HYDRO or HYD_VISC) and /EOS/POLYNOMIAL use this equation to compute hydrostatic pressure. It is possible to consider absolute values or relative variation. Material is supposed to be a perfect gas. The following cases have been investigated.
Table 2. Modeling formulation for perfect gas
Case Mathematical Model Pressure Energy
1 P ( μ , E ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGqbWaaeWaaeaacqaH8oqBcaGGSaGaamyraaGaayjkaiaawMca aaaa@3E65@ absolute absolute
2 Δ P ( μ , E ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqqHuoarcaWGqbWaaeWaaeaacqaH8oqBcaGGSaGaamyraaGaayjk aiaawMcaaaaa@3FCB@ relative absolute
3 Δ P ( μ , Δ E ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqqHuoarcaWGqbWaaeWaaeaacqaH8oqBcaGGSaGaeuiLdqKaamyr aaGaayjkaiaawMcaaaaa@4131@ relative relative
4 P ( μ , Δ E ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGqbWaaeWaaeaacqaH8oqBcaGGSaGaeuiLdqKaamyraaGaayjk aiaawMcaaaaa@3FCB@ absolute relative

Sound Speed and Time Step

Material LAW6 computes sound speed through the usual expression for fluids:(15) c 2 = d P d ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGJbWaaWbaaSqabeaacaaIYaaaaOGaeyypa0ZaaSaaaeaacaWG KbGaamiuaaqaaiaadsgacqaHbpGCaaaaaa@402F@
It can be written in function of μ :(16) μ = ρ ρ 0 1 1 d ρ = 1 ρ 0 1 d μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaH8oqBcqGH9aqpdaWcaaqaaiabeg8aYbqaaiabeg8aYnaaBaaa leaacaaIWaaabeaaaaGccqGHsislcaaIXaGaeyO0H49aaSaaaeaaca aIXaaabaGaamizaiabeg8aYbaacqGH9aqpdaWcaaqaaiaaigdaaeaa cqaHbpGCdaWgaaWcbaGaaGimaaqabaaaaOWaaSaaaeaacaaIXaaaba GaamizaiabeY7aTbaaaaa@4F77@
Then,(17) c 2 = 1 ρ 0 d P d μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGJbWaaWbaaSqabeaacaaIYaaaaOGaeyypa0ZaaSaaaeaacaaI XaaabaGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaaaakmaalaaabaGaam izaiaadcfaaeaacaWGKbGaeqiVd0gaaaaa@43A0@
The total differential of P in terms of internal energy E and μ is:(18) d P ( μ , E ) = P μ | E d μ + P E | μ d E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGKbGaamiuamaabmaabaGaeqiVd0MaaiilaiaadweaaiaawIca caGLPaaacqGH9aqpdaabcaqaamaalaaabaGaeyOaIyRaamiuaaqaai abgkGi2kabeY7aTbaaaiaawIa7amaaBaaaleaacaWGfbaabeaakiaa ysW7caWGKbGaeqiVd0Maey4kaSYaaqGaaeaadaWcaaqaaiabgkGi2k aadcfaaeaacqGHciITcaWGfbaaaaGaayjcSdWaaSbaaSqaaiabeY7a TbqabaGccaWGKbGaamyraaaa@570F@
In case of an isentropic transformation (reversible and adiabatic), the change of internal energy E int with volume V and pressure P is given by:(19) d E int = P d V
Using relation which links E int and E leads to:(20) d E = P V 0 d V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGKbGaamyraiabg2da9iabgkHiTmaalaaabaGaamiuaaqaaiaa dAfadaWgaaWcbaGaaGimaaqabaaaaOGaamizaiaadAfaaaa@40F1@
μ can be expressed in terms of volume ratio:(21) μ= v 0 v 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBcq GH9aqpdaWcaaqaaiaadAhadaWgaaWcbaGaaGimaaqabaaakeaacaWG 2baaaiabgkHiTiaaigdaaaa@3EC2@
Its variation in function of the volume change is also:(22) d μ = V 0 V 2 d V = ( 1 + μ ) 2 V 0 d V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGKbGaeqiVd0Maeyypa0JaeyOeI0YaaSaaaeaacaWGwbWaaSba aSqaaiaaicdaaeqaaaGcbaGaamOvamaaCaaaleqabaGaaGOmaaaaaa GccaWGKbGaamOvaiabg2da9iabgkHiTmaalaaabaWaaeWaaeaacaaI XaGaey4kaSIaeqiVd0gacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYa aaaaGcbaGaamOvamaaBaaaleaacaaIWaaabeaaaaGccaWGKbGaamOv aaaa@4E37@
Change in internal energy per unit volume E is then:(23) d E = P ( 1 + μ ) 2 d μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGKbGaamyraiabg2da9iabgkHiTmaalaaabaGaamiuaaqaamaa bmaabaGaaGymaiabgUcaRiabeY7aTbGaayjkaiaawMcaamaaCaaale qabaGaaGOmaaaaaaGccaWGKbGaeqiVd0gaaa@45D0@ (24) dP( μ,E ) dμ = P μ | E + P ( 1+μ ) 2 P E | μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai aadsgacaWGqbWaaeWaaeaacqaH8oqBcaGGSaGaamyraaGaayjkaiaa wMcaaaqaaiaadsgacqaH8oqBaaGaeyypa0ZaaqGaaeaadaWcaaqaai abgkGi2kaadcfaaeaacqGHciITcqaH8oqBaaaacaGLiWoadaWgaaWc baGaamyraaqabaGccqGHRaWkdaWcaaqaaiaadcfaaeaadaqadaqaai aaigdacqGHRaWkcqaH8oqBaiaawIcacaGLPaaadaahaaWcbeqaaiaa ikdaaaaaaOWaaqGaaeaadaWcaaqaaiabgkGi2kaadcfaaeaacqGHci ITcaWGfbaaaaGaayjcSdWaaSbaaSqaaiabeY7aTbqabaaaaa@591A@
Finally, the sound speed is given by:(25) c 2 = 1 ρ 0 P μ | E + P ρ 0 ( 1 + μ ) 2 P E | μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGJbWaaWbaaSqabeaacaaIYaaaaOGaeyypa0ZaaSaaaeaacaaI XaaabaGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaaaakmaaeiaabaWaaS aaaeaacqGHciITcaWGqbaabaGaeyOaIyRaeqiVd0gaaaGaayjcSdWa aSbaaSqaaiaadweaaeqaaOGaey4kaSYaaSaaaeaacaWGqbaabaGaeq yWdi3aaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacaaIXaGaey4kaSIa eqiVd0gacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaakmaaei aabaWaaSaaaeaacqGHciITcaWGqbaabaGaeyOaIyRaamyraaaaaiaa wIa7amaaBaaaleaacqaH8oqBaeqaaaaa@5969@
This expression computes the sound speed for a given equation of state P ( μ , E ) . In the case of perfect gas, it was shown that for each type of formulation (absolute or relative), EOS can be written:(26) P ( μ , E ) = C 0 + C 1 μ + ( C 4 + C 5 μ ) E
Equation 25 is used to compute sound speed:(27) P μ | E = C 1 + C 5 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaabcaqaamaalaaabaGaeyOaIyRaamiuaaqaaiabgkGi2kabeY7a TbaaaiaawIa7amaaBaaaleaacaWGfbaabeaakiabg2da9iaadoeada WgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaaiwda aeqaaOGaamyraaaa@46FC@ (28) P E | μ = C 4 + C 5 μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaabcaqaamaalaaabaGaeyOaIyRaamiuaaqaaiabgkGi2kaadwea aaaacaGLiWoadaWgaaWcbaGaeqiVd0gabeaakiabg2da9iaadoeada WgaaWcbaGaaGinaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaaiwda aeqaaOGaeqiVd0gaaa@47EB@ (29) c 2 = C 1 + C 5 E ρ 0 + C 0 + C 1 μ + ( C 4 + C 5 μ ) E ρ 0 ( 1 + μ ) 2 ( C 4 + C 5 μ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGJbWaaWbaaSqabeaacaaIYaaaaOGaeyypa0ZaaSaaaeaacaWG dbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaam4qamaaBaaaleaaca aI1aaabeaakiaadweaaeaacqaHbpGCcaaIWaaaamaaBaaaleaaaeqa aOGaey4kaSYaaSaaaeaacaWGdbWaaSbaaSqaaiaaicdaaeqaaOGaey 4kaSIaam4qamaaBaaaleaacaaIXaaabeaakiabeY7aTjabgUcaRmaa bmaabaGaam4qamaaBaaaleaacaaI0aaabeaakiabgUcaRiaadoeada WgaaWcbaGaaGynaaqabaGccqaH8oqBaiaawIcacaGLPaaacaWGfbaa baGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacaaIXaGaey 4kaSIaeqiVd0gacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa kmaabmaabaGaam4qamaaBaaaleaacaaI0aaabeaakiabgUcaRiaado eadaWgaaWcbaGaaGynaaqabaGccqaH8oqBaiaawIcacaGLPaaaaaa@63F4@
This calculation is then applied for each of the four cases.
Table 3. Numerical Sound Speed versus Theoretical Expression
Case C0 C1 C4 C5 c2

From Equation 25

Comparison with Theoretical Value
1 0 0 γ 1 γ 1 γ ( γ 1 ) E ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaWcaaqaaiabeo7aNnaabmaabaGaeq4SdCMaeyOeI0IaaGymaaGa ayjkaiaawMcaaiaadweaaeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba aaaaaa@42D6@ c= c REF MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbGaey ypa0Jaam4yamaaCaaaleqabaGaamOuaiaadweacaWGgbaaaaaa@3CD7@
2 0 0 γ 1 γ 1 γ ( γ 1 ) E ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaWcaaqaaiabeo7aNnaabmaabaGaeq4SdCMaeyOeI0IaaGymaaGa ayjkaiaawMcaaiaadweaaeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba aaaaaa@42D6@ c = c R E F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbGaey ypa0Jaam4yamaaCaaaleqabaGaamOuaiaadweacaWGgbaaaaaa@3CD7@
3 E 0 ( γ 1 ) E 0 ( γ 1 ) γ 1 γ 1 γ ( γ 1 ) ( E + E 0 ) ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaWcaaqaaiabeo7aNnaabmaabaGaeq4SdCMaeyOeI0IaaGymaaGa ayjkaiaawMcaamaabmaabaGaamyraiabgUcaRiaadweadaWgaaWcba GaaGimaaqabaaakiaawIcacaGLPaaaaeaacqaHbpGCdaWgaaWcbaGa aGimaaqabaaaaaaa@46FB@ c = c R E F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbGaey ypa0Jaam4yamaaCaaaleqabaGaamOuaiaadweacaWGgbaaaaaa@3CD7@
4 E 0 ( γ 1 ) E 0 ( γ 1 ) γ 1 γ 1 γ ( γ 1 ) ( E + E 0 ) ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaWcaaqaaiabeo7aNnaabmaabaGaeq4SdCMaeyOeI0IaaGymaaGa ayjkaiaawMcaamaabmaabaGaamyraiabgUcaRiaadweadaWgaaWcba GaaGimaaqabaaakiaawIcacaGLPaaaaeaacqaHbpGCdaWgaaWcbaGa aGimaaqabaaaaaaa@46FB@ c = c R E F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbGaey ypa0Jaam4yamaaCaaaleqabaGaamOuaiaadweacaWGgbaaaaaa@3CD7@

For each of the four formulations, the computed sound speed by Radioss is the same as the theoretical one. Time step and cycle number are also not affected.

Case 1: Both Pressure and Energy are Absolute Values

  1. Pressure:
    Equation of State:(30) P = ( γ 1 ) E int V = ( γ 1 ) ( 1 + μ ) E int V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0ZaaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGLOaGaayzkaaWa aSaaaeaacaWGfbWaaSbaaSqaaiGacMgacaGGUbGaaiiDaaqabaaake aacaWGwbaaaiabg2da9maabmaabaGaeq4SdCMaeyOeI0IaaGymaaGa ayjkaiaawMcaamaabmaabaGaaGymaiabgUcaRiabeY7aTbGaayjkai aawMcaamaalaaabaGaamyramaaBaaaleaaciGGPbGaaiOBaiaacsha aeqaaaGcbaGaamOvamaaBaaaleaacaaIWaaabeaaaaaaaa@5342@

    Where, E int | t = 0 = E 0 V 0 = P 0 V 0 γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaabcaqaai aadweadaWgaaWcbaGaciyAaiaac6gacaGG0baabeaaaOGaayjcSdWa aSbaaSqaaiaadshacqGH9aqpcaaIWaaabeaakiabg2da9iaadweada WgaaWcbaGaaGimaaqabaGccaWGwbWaaSbaaSqaaiaaicdaaeqaaOGa eyypa0ZaaSaaaeaacaWGqbWaaSbaaSqaaiaaicdaaeqaaOGaamOvam aaBaaaleaacaaIWaaabeaaaOqaaiabeo7aNjabgkHiTiaaigdaaaaa aa@4C45@

    Identifying the polynomial coefficients leads to:(31) P = ( C 4 + C 5 μ ) E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0ZaaeWaaeaacaWGdbWaaSbaaSqaaiaaisdaaeqaaOGaey4kaSIa am4qamaaBaaaleaacaaI1aaabeaakiabeY7aTbGaayjkaiaawMcaai aadweaaaa@41A6@

    Where, C 4 = C 5 = ( γ 1 ) ; E 0 = P 0 γ 1 ; P s h = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGdbWaaSbaaSqaaiaaisdaaeqaaOGaeyypa0Jaam4qamaaBeaa leaacaaI1aaabeaakiaaykW7cqGH9aqpdaqadaqaaiabeo7aNjabgk HiTiaaigdaaiaawIcacaGLPaaacaGGSaGaaGzbVlaadweadaWgaaWc baGaaGimaaqabaGccqGH9aqpdaWcaaqaaiaadcfadaWgaaWcbaGaaG imaaqabaaakeaacqaHZoWzcqGHsislcaaIXaaaaiaacYcacaaMf8Ua amiuamaaBaaaleaacaWGZbGaamiAaaqabaGccqGH9aqpcaaIWaaaaa@55D0@

  2. Corresponding Input:
    #---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
    /MAT/HYD_VISC/1
    Polynomial EOS-Absolute Pressure-Absolute Energy
    #              RHO_I               RHO_0
                   1.204                   0
    #                Knu                Pmin
                       0                   0
    /EOS/POLYNOMIAL/1
    Polynomial EOS-Absolute Pressure-Absolute Energy
    #                 C0                  C1                  C2                  C3
                       0                   0                   0                   0
    #                 C4                  C5                  E0                 Psh               RHO_0
                      .4                  .4              250000                   0               1.204
    #---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
  3. Output Results:
    Table 4.
    Time History Measure Initial Value Unit
    /TH/BRICK ( P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbaaaa@383D@ ) P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbaaaa@383D@ P 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS baaSqaaiaaicdaaeqaaaaa@3923@ Pressure
    /TH (IE) E int ( = E V 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS baaSqaaiGacMgacaGGUbGaaiiDaaqabaGcdaqadaqaaiabg2da9iaa dweacqGHflY1caWGwbWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaay zkaaaaaa@42AF@ E 0 V 0 Energy
    /TH/BRICK (IE) E int / V E 0 Pressure
  4. Comparison with Theoretical Result:
    Numerical result for perfect gas pressure is given by time history. Element time history (/TH/BRICK) allows displaying it. This result is compared to a theoretical one. Curves are superimposed.

    ex43_numerical_pressure_model1
    Figure 7. Numerical Pressure, Model 1
    Internal energy can be obtained through two different ways. The first one is internal energy density ( E int / V ) recorded by element time history (/TH/BRICK). The second one is the internal energy from the global time history e l e m e n t E int MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaaeqaqaai aadweadaWgaaWcbaGaciyAaiaac6gacaGG0baabeaaaeaacaWGLbGa amiBaiaadwgacaWGTbGaamyzaiaad6gacaWG0baabeqdcqGHris5aa aa@439C@ because the model is composed of a single element.

    ex43_numerical_internal_energy_model1
    Figure 8. Numerical Internal Energy, Model 1

Case 2: Pressure is Relative and Energy is Absolute

  1. Pressure:
    Equation of State:(32) P ( μ , E ) = ( γ 1 ) ( 1 + μ ) E int V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGqbWaaeWaaeaacqaH8oqBcaGGSaGaamyraaGaayjkaiaawMca aiabg2da9maabmaabaGaeq4SdCMaeyOeI0IaaGymaaGaayjkaiaawM caamaabmaabaGaaGymaiabgUcaRiabeY7aTbGaayjkaiaawMcaamaa laaabaGaamyramaaBaaaleaaciGGPbGaaiOBaiaacshaaeqaaaGcba GaamOvamaaBaaaleaacaaIWaaabeaaaaaaaa@4EC9@
    Relative Pressure:(33) Δ P = P ( μ , E ) P 0 = ( γ 1 ) ( 1 + μ ) E int V 0 P 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqqHuoarcaWGqbGaeyypa0JaamiuamaabmaabaGaeqiVd0Maaiil aiaadweaaiaawIcacaGLPaaacqGHsislcaWGqbWaaSbaaSqaaiaaic daaeqaaOGaeyypa0ZaaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGL OaGaayzkaaWaaeWaaeaacaaIXaGaey4kaSIaeqiVd0gacaGLOaGaay zkaaWaaSaaaeaacaWGfbWaaSbaaSqaaiGacMgacaGGUbGaaiiDaaqa baaakeaacaWGwbWaaSbaaSqaaiaaicdaaeqaaaaakiabgkHiTiaadc fadaWgaaWcbaGaaGimaaqabaaaaa@576E@
    Identifying with polynomial coefficients leads to:(34) Δ P = P = P s h = P s h + ( C 4 + C 5 μ ) E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHuoarca WGqbGaeyypa0Jaamiuaiabg2da9iaadcfadaWgaaWcbaGaam4Caiaa dIgaaeqaaOGaeyypa0JaeyOeI0IaamiuamaaBaaaleaacaWGZbGaam iAaaqabaGccqGHRaWkdaqadaqaaiaadoeadaWgaaWcbaGaaGinaaqa baGccqGHRaWkcaWGdbWaaSbaaSqaaiaaiwdaaeqaaOGaeqiVd0gaca GLOaGaayzkaaGaamyraaaa@4D9C@

    Where, C 4 = C 5 = ( γ 1 ) ; E 0 = P 0 γ 1 ; P s h = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGdbWaaSbaaSqaaiaaisdaaeqaaOGaeyypa0Jaam4qamaaBeaa leaacaaI1aaabeaakiaaykW7cqGH9aqpdaqadaqaaiabeo7aNjabgk HiTiaaigdaaiaawIcacaGLPaaacaGGSaGaaGzbVlaadweadaWgaaWc baGaaGimaaqabaGccqGH9aqpdaWcaaqaaiaadcfadaWgaaWcbaGaaG imaaqabaaakeaacqaHZoWzcqGHsislcaaIXaaaaiaacYcacaaMf8Ua amiuamaaBaaaleaacaWGZbGaamiAaaqabaGccqGH9aqpcaaIWaaaaa@55D0@

    Minimum Pressure:(35) P min = P 0

    Due to P > 0 Δ P P 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey Opa4JaaGimaiabgkDiElabfs5aejaadcfacqGHLjYScqGHsislcaWG qbWaaSbaaSqaaiaaicdaaeqaaaaa@4304@ , the minimum pressure must be set to a non-zero value.

  2. Corresponding Input:
    #---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
    /MAT/HYD_VISC/1
    Polynomial EOS-Relative Pressure-Absolute Energy
    #              RHO_I               RHO_0
                   1.204                   0
    #                Knu                Pmin
               1.5256E-5             -100000
    /EOS/POLYNOMIAL/1
    Polynomial EOS-Relative Pressure-Absolute Energy
    #                 C0                  C1                  C2                  C3
                       0                   0                   0                   0
    #                 C4                  C5                  E0                 Psh               RHO_0
                      .4                  .4              250000              100000               1.204
    #---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
  3. Output Result:
    Time History Measure Initial Value Unit
    /TH/BRICK ( P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbaaaa@383D@ ) Δ P 0 Pressure
    /TH (IE) E int ( = E V 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS baaSqaaiGacMgacaGGUbGaaiiDaaqabaGcdaqadaqaaiabg2da9iaa dweacqGHflY1caWGwbWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaay zkaaaaaa@42AF@ E 0 V 0 Energy
    /TH/BRICK (IE) E int / V E 0 Pressure
  4. Comparison with Theoretical Result:

    Numerical result for perfect gas pressure is given by time history. Element time history (/TH/BRICK) allows displaying it. This result is compared to a theoretical one. Curves are superimposed.

    Element time history (/TH/BRICK) is the pressure relative to Psh. The resulting curve is then shifted with Psh value and starts from 0.

    ex43_numerical_pressure_model2
    Figure 9. Numerical Pressure, Model 2
    Internal energy can be obtained through two different ways. The first one is internal energy density ( E int / V ) recorded by element time history (/TH/BRICK). The second one is the internal energy from the global time history e l e m e n t E int MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaaeqaqaai aadweadaWgaaWcbaGaciyAaiaac6gacaGG0baabeaaaeaacaWGLbGa amiBaiaadwgacaWGTbGaamyzaiaad6gacaWG0baabeqdcqGHris5aa aa@439C@ because the model is composed of a single element.

    ex43_numerical_internal_energy_model2
    Figure 10. Numerical Internal Energy, Model 2

Case 3: Both Pressure and Energy are Relative

  1. Pressure:
    Equation of State:(36) P = ( γ 1 ) ( 1 + μ ) E int V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0ZaaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGLOaGaayzkaaWa aeWaaeaacaaIXaGaey4kaSIaeqiVd0gacaGLOaGaayzkaaWaaSaaae aacaWGfbWaaSbaaSqaaiGacMgacaGGUbGaaiiDaaqabaaakeaacaWG wbWaaSbaaSqaaiaaicdaaeqaaaaaaaa@48A0@
    Initial internal energy can be introduced:(37) E int = E int + ( E int | t = 0 E int | t = 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS baaSqaaiGacMgacaGGUbGaaiiDaaqabaGccqGH9aqpcaWGfbWaaSba aSqaaiGacMgacaGGUbGaaiiDaaqabaGccqGHRaWkdaqadaqaamaaei aabaGaamyramaaBaaaleaaciGGPbGaaiOBaiaacshaaeqaaaGccaGL iWoadaWgaaWcbaGaamiDaiabg2da9iaaicdaaeqaaOGaeyOeI0Yaaq GaaeaacaWGfbWaaSbaaSqaaiGacMgacaGGUbGaaiiDaaqabaaakiaa wIa7amaaBaaaleaacaWG0bGaeyypa0JaaGimaaqabaaakiaawIcaca GLPaaaaaa@5433@
    Pressure from a reference one provides:(38) P P 0 = Δ P = ( γ 1 ) ( 1 + μ ) ( Δ E + E 0 ) P 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey OeI0IaamiuamaaBaaaleaacaaIWaaabeaakiabg2da9iabfs5aejaa dcfacqGH9aqpdaqadaqaaiabeo7aNjabgkHiTiaaigdaaiaawIcaca GLPaaadaqadaqaaiaaigdacqGHRaWkcqaH8oqBaiaawIcacaGLPaaa daqadaqaaiabfs5aejaadweacqGHRaWkcaWGfbWaaSbaaSqaaiaaic daaeqaaaGccaGLOaGaayzkaaGaeyOeI0IaamiuamaaBaaaleaacaaI Waaabeaaaaa@51E6@

    Where, Δ E = E int E int | t = 0 V 0 ; E 0 = E int | t = 0 V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHuoarca WGfbGaeyypa0ZaaSaaaeaacaWGfbWaaSbaaSqaaiGacMgacaGGUbGa aiiDaaqabaGccqGHsisldaabcaqaaiaadweadaWgaaWcbaGaciyAai aac6gacaGG0baabeaaaOGaayjcSdWaaSbaaSqaaiaadshacqGH9aqp caaIWaaabeaaaOqaaiaadAfadaWgaaWcbaGaaGimaaqabaaaaOGaai 4oaiaaysW7caWGfbWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0ZaaSaa aeaadaabcaqaaiaadweadaWgaaWcbaGaciyAaiaac6gacaGG0baabe aaaOGaayjcSdWaaSbaaSqaaiaadshacqGH9aqpcaaIWaaabeaaaOqa aiaadAfadaWgaaWcbaGaaGimaaqabaaaaaaa@58D7@ .

    Identifying with polynomial coefficients leads to:(39) Δ P = P P s h = C 0 + C 1 μ + ( C 4 + C 5 μ ) Δ E P s h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHuoarca WGqbGaeyypa0JaamiuaiabgkHiTiaadcfadaWgaaWcbaGaam4Caiaa dIgaaeqaaOGaeyypa0Jaam4qamaaBaaaleaacaaIWaaabeaakiabgU caRiaadoeadaWgaaWcbaGaaGymaaqabaGccqaH8oqBcqGHRaWkdaqa daqaaiaadoeadaWgaaWcbaGaaGinaaqabaGccqGHRaWkcaWGdbWaaS baaSqaaiaaiwdaaeqaaOGaeqiVd0gacaGLOaGaayzkaaGaeuiLdqKa amyraiabgkHiTiaadcfadaWgaaWcbaGaam4CaiaadIgaaeqaaaaa@54E8@

    Where, C 0 = C 1 = E 0 ( γ 1 ) , C 4 = C 5 = γ 1 , Δ E 0 = 0 and P s h = P 0 .

    Minimum Pressure:(40) P min = P 0

    Due to P 0 Δ P P 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey yzImRaaGimaiabgkDiElabfs5aejaadcfacqGHLjYScqGHsislcaWG qbWaaSbaaSqaaiaaicdaaeqaaaaa@43C2@ , the minimum pressure must be set to a non-zero value.

  2. Corresponding Input:
    #---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
    /MAT/HYD_VISC/1
    Polynomial EOS-Relative Pressure-Absolute Energy
    #              RHO_I               RHO_0
                   1.204                   0
    #                Knu                Pmin
               1.5256E-5             -100000
    /EOS/POLYNOMIAL/1
    Polynomial EOS-Relative Pressure-Absolute Energy
    #                 C0                  C1                  C2                  C3
                  100000              100000                   0                   0
    #                 C4                  C5                  E0                 Psh               RHO_0
                      .4                  .4                   0              100000               1.204
    #---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
  3. Output Results:
    Time History Measure Initial Value Unit
    /TH/BRICK ( P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbaaaa@383D@ ) Δ P 0 Pressure
    /TH (IE) E int ( = Δ E V 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS baaSqaaiGacMgacaGGUbGaaiiDaaqabaGcdaqadaqaaiabg2da9iab fs5aejaadweacqGHflY1caWGwbWaaSbaaSqaaiaaicdaaeqaaaGcca GLOaGaayzkaaaaaa@4415@ 0 Energy
    /TH/BRICK (IE) Δ E int / V 0 Pressure
  4. Comparison with Theoretical Result:

    Numerical result for perfect gas pressure is given by time history. Element time history (/TH/BRICK) allows displaying it. This result is compared to a theoretical one. Curves are superimposed.

    Element time history (/TH/BRICK) is the pressure relative to Psh. The resulting curve is then shifted with Psh value and starts also from 0.

    ex43_numerical_pressure_model3
    Figure 11. Numerical Pressure, Model 3
    Internal energy can be obtained through two different ways. The first one is internal energy density ( E int / V ) recorded by element time history (/TH/BRICK). The second one is the internal energy from the global time history e l e m e n t E int MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaaeqaqaai aadweadaWgaaWcbaGaciyAaiaac6gacaGG0baabeaaaeaacaWGLbGa amiBaiaadwgacaWGTbGaamyzaiaad6gacaWG0baabeqdcqGHris5aa aa@439C@ because the model is composed of a single element. This numerical internal energy is relative to its initial value; it is shifted with the E 0 V 0 value from the absolute theoretical one and also starts from 0.

    ex43_numerical_internal_energy_model3
    Figure 12. Numerical Internal Energy, Model 3

Case 4: Pressure is Absolute and Energy is Relative

  1. Pressure:
    Equation of State:(41) P = ( γ 1 ) ( 1 + μ ) E int V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0ZaaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGLOaGaayzkaaWa aeWaaeaacaaIXaGaey4kaSIaeqiVd0gacaGLOaGaayzkaaWaaSaaae aacaWGfbWaaSbaaSqaaiGacMgacaGGUbGaaiiDaaqabaaakeaacaWG wbWaaSbaaSqaaiaaicdaaeqaaaaaaaa@48A0@
    Initial internal energy can be introduced:(42) E int = E int + ( E int | t = 0 E int | t = 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS baaSqaaiGacMgacaGGUbGaaiiDaaqabaGccqGH9aqpcaWGfbWaaSba aSqaaiGacMgacaGGUbGaaiiDaaqabaGccqGHRaWkdaqadaqaamaaei aabaGaamyramaaBaaaleaaciGGPbGaaiOBaiaacshaaeqaaaGccaGL iWoadaWgaaWcbaGaamiDaiabg2da9iaaicdaaeqaaOGaeyOeI0Yaaq GaaeaacaWGfbWaaSbaaSqaaiGacMgacaGGUbGaaiiDaaqabaaakiaa wIa7amaaBaaaleaacaWG0bGaeyypa0JaaGimaaqabaaakiaawIcaca GLPaaaaaa@5433@
    Pressure from a reference provided:(43) P = ( γ 1 ) ( 1 + μ ) ( Δ E + E 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0ZaaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGLOaGaayzkaaWa aeWaaeaacaaIXaGaey4kaSIaeqiVd0gacaGLOaGaayzkaaWaaeWaae aacqqHuoarcaWGfbGaey4kaSIaamyramaaBaaaleaacaaIWaaabeaa aOGaayjkaiaawMcaaaaa@494B@
    Identifying with polynomial coefficients leads to:(44) P = C 0 + C 1 μ + ( C 4 + C 5 μ ) Δ E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0Jaam4qamaaBaaaleaacaaIWaaabeaakiabgUcaRiaadoeadaWg aaWcbaGaaGymaaqabaGccqaH8oqBcqGHRaWkdaqadaqaaiaadoeada WgaaWcbaGaaGinaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaaiwda aeqaaOGaeqiVd0gacaGLOaGaayzkaaGaeuiLdqKaamyraaaa@49F7@

    Where, C 0 = C 1 = E 0 ( γ 1 ) ; C 4 = C 5 = γ 1

  2. Corresponding Input:
    #---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
    /MAT/HYD_VISC/1
    Polynomial EOS-Relative Pressure-Absolute Energy
    #              RHO_I               RHO_0
                   1.204                   0
    #                Knu                Pmin
               1.5256E-5                   0
    /EOS/POLYNOMIAL/1
    Polynomial EOS-Relative Pressure-Absolute Energy
    #                 C0                  C1                  C2                  C3
                  100000              100000                   0                   0
    #                 C4                  C5                  E0                 Psh               RHO_0
                      .4                  .4                   0                   0               1.204
    #---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
  3. Output Results:
    Time History Measure Initial Value Unit
    /TH/BRICK ( P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbaaaa@383D@ ) P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbaaaa@383D@ P 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS baaSqaaiaaicdaaeqaaaaa@3923@ Pressure
    /TH (IE) E int ( = Δ E V 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS baaSqaaiGacMgacaGGUbGaaiiDaaqabaGcdaqadaqaaiabg2da9iab fs5aejaadweacqGHflY1caWGwbWaaSbaaSqaaiaaicdaaeqaaaGcca GLOaGaayzkaaaaaa@4415@ 0 Energy
    /TH/BRICK (IE) Δ E int / V 0 Pressure
  4. Comparison with Theoretical Result:
    Element time history (/TH/BRICK) gives absolute pressure. This result is compared to a theoretical one. Curves are superimposed.

    ex43_numerical_pressure_model4
    Figure 13. Numerical Pressure, Model 4
    Internal energy can be obtained through two different ways. The first one is internal energy density ( Δ E int / V ) recorded by element time history (/TH/BRICK). The second one is the internal energy from the global time history e l e m e n t E int MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaaeqaqaai aadweadaWgaaWcbaGaciyAaiaac6gacaGG0baabeaaaeaacaWGLbGa amiBaiaadwgacaWGTbGaamyzaiaad6gacaWG0baabeqdcqGHris5aa aa@439C@ because the model is composed of a single element. This numerical internal energy is relative to its initial value; it is shifted with the E 0 V 0 value from the absolute theoretical one and also starts from 0.

    ex43_numerical_internal_energy_model4
    Figure 14. Numerical Internal Energy, Model 4