Managan, R. A.

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LLNL-PROC-614554Using Tabular EOS at LowTemperaturesR. A. ManaganJanuary 31, 2013Nuclear Explosives Code Developers ConferenceLivermore, CA, United StatesOctober 22, 2012 through October 26, 2012Disclaimer  This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.  LLNL-PRES-xxxx UNCLASSIFIED July 16, 2012Using Tabular EOS at Low Temperatures (U)1Robert A. Managan1Lawrence Livermore National Laboratory, Livermore, CAAbstractIn multi-physics hydro-codes there are many algorithms that update the internal energy of a zone. It is thejob of the equation of state to take this new internal energy and calculate a new temperature. When thephysics of the problem requires that the code use separate electron and ion temperatures problems can arisewhen determining the new temperature. This is due to the fact that in the LEOS tables the electron energyis zero until there are free electrons while the algorithms that couple energy into the electrons generallyassume a plasma and don’t account for whether the electrons are bound or free.A new approach for dealing with these issues is presented that avoids discontinuous jumps in the electrontemperature. It does this by smoothly switching from a total energy solution to separate temperature updatesfor electrons and ions. (U)IntroductionThe Livermore Equation of State (LEOS) tables are built with separate tables for the cold, electron, andion values. These three tables are also combined into a table for the total values. When a multi-physicscode is run in a mode that has separate electron and ion temperatures we need to use the separate tables forthe EOS. However, most multi–physics codes are based on algorithms that are derived in the plasmaphysics regime and make different assumptions in the low temperature limit than the EOS tables do. Inparticular the table’s electron energy is zero at low temperatures until there are free electrons, but theradiation–electron coupling transfers energy to the electrons without ionizing them. The result is that oncea little energy is put in to the electron field the electron temperature jumps up only to drop the next cyclewhen the electron energy is coupled to the ions.The original algorithm for updating the temperature is to iterate on the temperature until the energy welook up matches the energy of the zone. These equations are used in every iteration to solve for the updatesto the electron and ion temperatures:Ee = E′e(Te) +∂E′e∂T∆Te (1)NECDC 2012 Proceedings UNCLASSIFIEDLLNL-PRES-xxxx UNCLASSIFIED July 16, 2012Ei = E′i(Ti) +∂E′i∂T∆Ti (2)where E is the zone’s internal energy we want to match, E′ is the energy evaluated at the current guess forthe temperature, and ∆T is the update to the temperature. The iteration proceeds until ∆T becomes smallenough.Take an ignition NIF capsule as an example. It starts at cryogenic temperatures but the simulation is runwith separate electron and ion temperatures since that is required when the fuel is burning. Fig. 1 showsthe electron temperature for a point in the outer half of the ablator. As the radiation heats the electrons thetemperature jumps up to temperatures between 6× 10−7 to 1× 10−5 keV and then gets pulled back downby electron-ion coupling. Clearly the simple iteration defined by equations 1 and 2 is inadequate and needsto be improved.ares Time = 9.14768e-03Cycle = 1400 Dtc:0(1,413) = 5.76999e-061.0e-06 1.0e-05 1.0e-04 1.0e-03 1.0e-025.0e-081.0e-075.0e-071.0e-065.0e-061.0e-055.0e-051.0e-045.0e-041.0e-030 tmat:0(1,284) vs. time  1.000e-10  9.148e-03  1.070e-07  2.208e-031 tion:0(1,284) vs. time  1.000e-10  9.148e-03  9.991e-07  2.207e-030 0000 00001 1 1 111111Figure 1: Electron temperature in the outer ablator using the old algorithm.The AlgorithmThe new method uses an interpolating function f(T ) = 2−(T/T0)2to switch from iterating on the totalenergy to iterating on the separate electron and ion energies. Currently T0, the “half-way” temperature, ischosen to be 5× 10−3 keV a temperature at which all materials have begun to ionize and have non-zeroNECDC 2012 Proceedings UNCLASSIFIEDLLNL-PRES-xxxx UNCLASSIFIED July 16, 2012electron energies. Different forms for the interpolating function or different values of T0 could be used butthis choice appears to be adequate.The temperature updates are calculated using these effective energies:E∗e =Ee + f(Te) (Ei + Ec)1 + f(Te)(3)E∗i =f(Te)Ee + (Ei + Ec)1 + f(Te)(4)At low temperatures f(Te) ' 1 so E∗e ' E∗i and both are equal to the half of the total internal energy. Athigh temperatures they are equal to the electron and ion energies respectively, where the ion energyincludes the cold energy. For updating temperatures we consider E∗e as a function only of Te and E∗i as afunction only of Ti. This results in using these equations to solve for ∆Te and ∆Ti:E∗e =E′e + f(Te) (E′i + Ec)1 + f(Te)+11 + f(Te){∂E′e∂Te+dfdTe[E′i + Ec −E′e + f (E′i + Ec)1 + f(Te)]+ f(Te)∂E′i∂Ti}∆Te (5)E∗i =f(Te)E′e + E′i + Ec1 + f(Te)+11 + f(Te)[f(Te)∂E′e∂Te+dfdTe(E′e −fE′e + E′i + Ec1 + f(Te))+∂E′i∂Ti]∆Ti (6)Using equations 3 and 4 we can rewrite these as:Ee − E′e + f(Te) (Ei − E′i)1 + f=11 + f{∂E′e∂Te+dfdTe[E′i + Ec −E′e + f (E′i + Ec)1 + f]+ f∂E′i∂Ti}∆Tef(Te)(Ee − E′e) + Ei − E′i1 + f=11 + f[f∂E′e∂Te+dfdTe(E′e −fE′e + E′i + Ec1 + f)+∂E′i∂Ti]∆Tiwhich can be simplified to:Ee − E′e + f(Te) (Ei − E′i)1 + f(Te)=11 + f(Te)[∂E′e∂Te− dfdTeE′e − (E′i + Ec)1 + f(Te)+ f(Te)∂E′i∂Ti]∆Te (7)f(Te)(Ee − E′e) + Ei − E′i1 + f(Te)=11 + f(Te)[f(Te)∂E′e∂Te+dfdTeE′e − (E′i + Ec)1 + f(Te)+∂E′i∂Ti]∆Ti (8)After each iteration E′e and E′i are evaluated using the current estimate of Te and Ti respectively. The righthand sides of equations 7 and 8 are effective electron and ion heat capacities. These are not used elsewherein the code.The sum of Eqs. 5 and 6 gives the total energy and guarantees that as the ∆T ’s go to zero the total energyis preserved. Of course, when f(T )→ 0 for large temperatures, the temperature updates are independentNECDC 2012 Proceedings UNCLASSIFIEDLLNL-PRES-xxxx UNCLASSIFIED July 16, 2012and the same as before. When f(t)→ 1 for small temperatures, the two temperature updates are the sameand the electron and ion temperatures move up or down together preserving any difference there isbetween them.When we apply this method for updating the temperature to the same NIF capsule as in Fig. 1 we get theresult in Fig. 2. The oscillations are gone and while a small temperature difference shows up it is neverlarge. Similar results are shown in figures 3 and 4 for a point in the inner half of the ablator.ares Time = 9.14766e-03Cycle = 1400 Dtc:0(1,413) = 5.77002e-061.0e-06 1.0e-05 1.0e-04 1.0e-03 1.0e-021.0e-065.0e-061.0e-055.0e-051.0e-045.0e-041.0e-030 tmat:0(1,284) vs. time  1.000e-10  9.148e-03  1.000e-06  2.208e-031 tion:0(1,284) vs. time  1.000e-10  9.148e-03  1.000e-06  2.207e-030 0 0 0000001 1 1 111111Figure 2: Electron temperature in the outer ablator using the new algorithm.What did not workMathematically the updates for E∗e and E∗i should each depend on both ∆Te and ∆Ti. This gives a twoequation system to be solved for the temperature updates ∆Te and ∆Ti.E∗e =E′e + f(Te) (E′i + Ec)1 + f+f1 + f∂Ei∂Ti∆Ti +11 + f{∂Ee∂Te+dfdTe[E′i + Ec −E′e + f (E′i + Ec)1 + f]}∆Te (9)NECDC 2012 Proceedings UNCLASSIFIEDLLNL-PRES-xxxx UNCLASSIFIED July 16, 2012ares Time = 9.14768e-03Cycle = 1400 Dtc:0(1,413) = 5.76999e-061.0e-06 1.0e-05 1.0e-04 1.0e-03 1.0e-025.0e-081.0e-075.0e-071.0e-065.0e-061.0e-055.0e-051.0e-045.0e-040 tmat:0(1,173) vs. time  1.000e-10  9.148e-03  1.070e-07  6.221e-041 tion:0(1,173) vs. time  1.000e-10  9.148e-03  9.907e-07  6.218e-040 0 000 0 0001 1 1 1 1 1111Figure 3: Electron temperature in the innerablator using the old algorithm.ares Time = 9.14766e-03Cycle = 1400 Dtc:0(1,413) = 5.77002e-061.0e-06 1.0e-05 1.0e-04 1.0e-03 1.0e-021.0e-065.0e-061.0e-055.0e-051.0e-045.0e-040 tmat:0(1,173) vs. time  1.000e-10  9.148e-03  1.000e-06  6.221e-041 tion:0(1,173) vs. time  1.000e-10  9.148e-03  1.000e-06  6.217e-040 0 0 0 0 00001 1 1 1 1 1111Figure 4: Electron temperature in the innerablator using the new algorithm.E∗i =f(Te)E′e + E′i + Ec1 + f+11 + f∂Ei∂Ti∆Ti +11 + f[f(Te)∂Ee∂Te+dfdTe(E′e −fE′e + E′i + Ec1 + f)]∆Te (10)Using equations 3 and 4 and simplifying, equations 9 and 10 can be rewritten as:Ee − E′e + f(Te) (Ei − E′i)1 + f=f1 + f∂Ei∂Ti∆Ti +11 + f{∂Ee∂Te+dfdTeE′i + Ec − E′e1 + f}∆Te (11)f(Te)(Ee − E′e) + Ei − E′i1 + f=11 + f∂Ei∂Ti∆Ti +11 + f[f(Te)∂Ee∂Te+dfdTeE′e − E′i − Ec1 + f]∆Te (12)This version of the algorithm suffers from an instability similar to the original algorithm since ∂Ee∂Te = 0 atlow temperatures when Ee ≡ 0 because there are no free electrons. When coupled with the fact thatdfdTe→ 0 at low temperatures the coefficient of ∆TE → 0 leaving the system of equations ill defined.ConclusionA new method for updating the temperature from the internal energy when there are separate electron andNECDC 2012 Proceedings UNCLASSIFIEDUNCLASSIFIEDion temperatures has been described. At low temperatures this method uses a total energy update thatapplies equal increments to the electron and ion temperatures. At high temperatures the electron and iontemperatures are updated independently based on the electron and ion internal energies. The methodswitches between these two limits with a smooth interpolation function. Using this method has removedone source of numerical noise from calculations of NIF implosions.NECDC 2012 Proceedings UNCLASSIFIED

Using Tabular EOS at Low Temperatures

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Using Tabular EOS at Low Temperatures

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