U3Si2Sifgrs

Computes fission gas release and swelling in U3Si2 through a physically-based model.

Description

The processes induced by the generation of the fission gases xenon and krypton in nuclear fuel have a strong impact on the thermo-mechanical performance of the fuel rods. On the one hand, the fission gases tend to precipitate into bubbles resulting in fuel swelling, which promotes pellet-cladding gap closure and the ensuing pellet-cladding mechanical interaction (PCMI). On the other hand, fission gas release (FGR) to the fuel rod free volume causes pressure build-up and thermal conductivity degradation of the rod filling gas.

The fundamental physical processes, which control the kinetics of fission gas swelling and release in irradiated fuel, may be summarized as follows; fission gas atoms generated in the fuel grains diffuse towards the grain boundaries through repeated trapping in and irradiation-induced resolution from nanometre-size intra-granular gas bubbles. Although a part of the gas atoms that reach the grain boundaries is dissolved back to the grain interior by irradiation, the majority of the gas diffuses into grain-face gas bubbles, giving rise to grain-face swelling. Bubble growth brings about bubble coalescence and inter-connection, eventually leading to the formation of a tunnel network through which a fraction of the gas is released to the fuel rod free volume.

In BISON, fission gas behavior is computed for each integration point in the fuel finite element mesh. The gas produced at each integration point is computed by a numerical time integration of the gas production rate, given as the product of the fission rate and fractional yield of gas atoms per fission.

Coupled fission gas release and swelling model

The U3Si2Sifgrs model handles fission gas swelling and release in U $var element = document.getElementById("moose-equation-d6be97ab-8f46-4107-9d3e-0ba9412707dc");katex.render("_3", element, {displayMode:false,throwOnError:false});$ Si $var element = document.getElementById("moose-equation-474ab540-712c-4a4c-a8ad-773b067d03d1");katex.render("_2", element, {displayMode:false,throwOnError:false});$ under power reactor conditions. The model calculates the coupled fission gas swelling and release concurrently and is physically based. This model relies on the current understanding of microstructure and fission gas behavior in U $var element = document.getElementById("moose-equation-1ed6bbcf-1cdd-453e-8ad6-6b41fad7ef4c");katex.render("_3", element, {displayMode:false,throwOnError:false});$ Si $var element = document.getElementById("moose-equation-9db3dc8c-2f82-42db-b702-3a07c37f5624");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , including the recent findings from lower-length scale modeling and the available experimental data. Based on the experimental evidence from Shimizu (1965), we assume U $var element = document.getElementById("moose-equation-7e4f8c36-ae3c-4134-99d0-4cc2eab6602c");katex.render("_3", element, {displayMode:false,throwOnError:false});$ Si $var element = document.getElementById("moose-equation-045ca335-abd0-4009-88f8-ce737abdd2bf");katex.render("_2", element, {displayMode:false,throwOnError:false});$ remains crystalline at power reactor temperatures. We also assume both intra-granular and grain-boundary gas bubbles develop, as in UO $var element = document.getElementById("moose-equation-9cd2e808-025f-4f22-a506-6683bda6b66d");katex.render("_2", element, {displayMode:false,throwOnError:false});$ . Accordingly, the same basic structure of the BISON UO $var element = document.getElementById("moose-equation-44cb46c8-6531-4bd7-93c7-70a978a52f00");katex.render("_2", element, {displayMode:false,throwOnError:false});$ ( UO2Sifgrs) model is applied in this model, with a two-stage description of intra-granular and inter-granular processes. However, the physical interpretation for some processes differs from the interpretation in the UO $var element = document.getElementById("moose-equation-e78f8441-8d39-4fed-8ad5-4afc4d81f949");katex.render("_2", element, {displayMode:false,throwOnError:false});$ model to better conform to the current understanding of fission gas behavior in U $var element = document.getElementById("moose-equation-5883d972-4da5-4ee0-9f66-50ca1aa6c847");katex.render("_3", element, {displayMode:false,throwOnError:false});$ Si $var element = document.getElementById("moose-equation-43f7f8c2-2193-4b9f-9803-36f1428e58f7");katex.render("_2", element, {displayMode:false,throwOnError:false});$ . In particular, two modeling aspects are included in U3Si2Sifgrs that are qualitatively different from the approach in the UO $var element = document.getElementById("moose-equation-6524c6d1-0206-446b-a493-70731dd4eb49");katex.render("_2", element, {displayMode:false,throwOnError:false});$ UO2Sifgrs model and were developed specifically for the U $var element = document.getElementById("moose-equation-d04ce2a3-c2ec-4283-a272-eed60b4d3ede");katex.render("_3", element, {displayMode:false,throwOnError:false});$ Si $var element = document.getElementById("moose-equation-e324c27a-1725-483a-aaae-3d0f3f128891");katex.render("_2", element, {displayMode:false,throwOnError:false});$ model, i.e., (1) modeling intra-granular bubble nucleation and re-solution based on the so-called homogeneous mechanisms and (2) an intra-granular bubble growth model that considers absorption of vacancies by the bubbles and is based on an adaptation of the Speight-Beere model.

In order to fill the experimental data gap for fission gas behavior in U $var element = document.getElementById("moose-equation-1476b847-3d56-4166-8f11-f72e849b5a07");katex.render("_3", element, {displayMode:false,throwOnError:false});$ Si $var element = document.getElementById("moose-equation-98717e86-8de7-48c4-b73e-9d6d72b1ea92");katex.render("_2", element, {displayMode:false,throwOnError:false});$ under power reactor conditions, a multiscale approach is adopted, whereby lower-length scale modeling for the parameters is used to inform the engineering scale calculation of U3Si2Sifgrs. In particular, we use (i) for the diffusion coefficients of gas atoms and vacancies, values from density functional theory calculations in Andersson (2017), (ii) for the re-solution rate of gas atoms from intra-granular bubbles, values from binary collision approximation calculations in Matthews et al. (2016), and (iii) for the U $var element = document.getElementById("moose-equation-caeca573-296a-48ef-aa81-efbe90233b78");katex.render("_3", element, {displayMode:false,throwOnError:false});$ Si $var element = document.getElementById("moose-equation-05443ccc-03f9-48e9-8737-9e5dbaf02665");katex.render("_2", element, {displayMode:false,throwOnError:false});$ -gas surface energy and the semidihedral angle of grain-boundary gas bubbles, values from molecular dynamics calculations in Beeler et al. (2019).

Parameters for which specific U $var element = document.getElementById("moose-equation-07dcb9de-2910-4a8c-a6fe-4198cebc821f");katex.render("_3", element, {displayMode:false,throwOnError:false});$ Si $var element = document.getElementById("moose-equation-a2e2458d-c7e7-4eb6-a22e-8b4c3ef275b7");katex.render("_2", element, {displayMode:false,throwOnError:false});$ values are not yet available are estimated based on data for metals, theoretical considerations or the best fitting of model results to experimental data. The U3Si2Sifgrs model will be progressively updated as new data become available.

Intra-granular gas behavior

The intra-granular module computes gas generation and bubble evolution in the grains and the coupled diffusion of gas atoms to grain boundaries. The description includes bubble nucleation, irradiation-induced re-solution of gas atoms from bubbles back into the lattice, absorption of gas atoms and vacancies at the bubbles, and diffusion of single gas atoms.

Nucleation and re-solution may occur by different mechanisms, i.e., heterogeneous and homogeneous (Olander and Wongsawaeng, 2006). Heterogeneous nucleation and re-solution refer to the creation of new bubbles nuclei as a direct consequence of the interaction of fission fragments with the lattice and the bubbles destruction occurring en-bloc by passing fission fragments, respectively. The homogeneous mechanisms accounts for the nucleation of bubbles by diffusion-driven interactions of dissolved gas atoms and re-solution occurring gradually by ejection of individual atoms. The dominant mechanisms depend upon the nature of the interactions between fission fragments and lattice (electronic or phononic). Based on Matthews et al. (2016), we assume the homogeneous mechanisms to dominate in U $var element = document.getElementById("moose-equation-94fa16d1-902f-4f57-99e5-1472ede64f68");katex.render("_3", element, {displayMode:false,throwOnError:false});$ Si $var element = document.getElementById("moose-equation-2ca1f21b-eee3-43b7-ba49-ecf27014d6ab");katex.render("_2", element, {displayMode:false,throwOnError:false});$ . The equations for the evolution of the intra-granular gas bubble number density and gas atom concentrations are: $(1)var element = document.getElementById("moose-equation-e3a589a3-405f-41c6-b5bf-8858a6e4e5b0");katex.render(" \\frac{dN}{dt} = \\nu - \\frac{b}{n-1} N", element, {displayMode:true,throwOnError:false});$

$(2)var element = document.getElementById("moose-equation-098bfd61-301b-4eb6-8afb-1fa77ea04bbc");katex.render(" \\frac{\\partial c}{\\partial t} = D \\frac{1}{r^2} \\frac{\\partial}{\\partial r} \\left( r^2 \\frac{\\partial c}{\\partial r} \\right) - g c + b m - 2 \\nu + \\beta", element, {displayMode:true,throwOnError:false});$

$(3)var element = document.getElementById("moose-equation-ec1b97c6-1da4-4c11-847d-64e07224a1bb");katex.render(" \\frac{\\partial m}{\\partial t} = + g c - b m + 2 \\nu", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-0baa9c0c-513f-41bf-97e6-800f6657a19f");katex.render("N", element, {displayMode:false,throwOnError:false});$ (m $var element = document.getElementById("moose-equation-e9f54c4a-e178-4096-9737-7674f1200ca0");katex.render("^{-3}", element, {displayMode:false,throwOnError:false});$ ) is the number density of intra-granular bubbles, $var element = document.getElementById("moose-equation-cfdce91d-6bed-4400-853d-20115f29402b");katex.render("n", element, {displayMode:false,throwOnError:false});$ is the number of gas atoms per bubble, $var element = document.getElementById("moose-equation-d99c168b-91e7-4d0d-85fa-24cce55ecafe");katex.render("c", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-964a5fe3-5764-4655-9aa5-36b6575c712a");katex.render("m", element, {displayMode:false,throwOnError:false});$ (m $var element = document.getElementById("moose-equation-a04b750f-09b8-42b0-a2da-ee0b4889c076");katex.render("^{-3}", element, {displayMode:false,throwOnError:false});$ ) are the intra-granular gas concentration in the matrix and in the bubbles, respectively, $var element = document.getElementById("moose-equation-4a13e44c-d73a-407c-b08b-600b9be616b2");katex.render("t", element, {displayMode:false,throwOnError:false});$ (s) the time, $var element = document.getElementById("moose-equation-96396bbd-bbf7-49a8-acaa-541ac05dee16");katex.render("D", element, {displayMode:false,throwOnError:false});$ (m $var element = document.getElementById("moose-equation-8441ea16-b11a-42d9-a4b5-279910db3424");katex.render("^2", element, {displayMode:false,throwOnError:false});$ s $var element = document.getElementById("moose-equation-4ec8ab99-3065-4218-ae94-8cc7bae8c5a6");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) the single-atom gas diffusion coefficient, $var element = document.getElementById("moose-equation-eaa5cee5-ce0b-4d42-99b1-57c57ca57d52");katex.render("r", element, {displayMode:false,throwOnError:false});$ (m) the radial coordinate in the spherical grain, $var element = document.getElementById("moose-equation-4adac032-a8e2-4bc0-9203-af8c9a27e162");katex.render("\\beta", element, {displayMode:false,throwOnError:false});$ (m $var element = document.getElementById("moose-equation-e76e6b70-0c85-47b4-bcde-d7e84606299d");katex.render("^{-3}", element, {displayMode:false,throwOnError:false});$ s $var element = document.getElementById("moose-equation-f1013052-5c18-413d-8a43-14d88acc3b2c");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) the gas generation rate, $var element = document.getElementById("moose-equation-f28136f4-463e-4c94-b0cc-f4efc4ef60ce");katex.render("g", element, {displayMode:false,throwOnError:false});$ (s $var element = document.getElementById("moose-equation-a473c46c-9f21-4588-aefc-7b2a53b22d6c");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) the trapping rate, $var element = document.getElementById("moose-equation-af8b9cf5-d595-4f2c-88d0-6d6f3363fdac");katex.render("b", element, {displayMode:false,throwOnError:false});$ (s $var element = document.getElementById("moose-equation-cf419e19-0e43-4ff2-91af-c6a763528baf");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) the re-solution rate. The coefficient of 2 for the nucleation rate $var element = document.getElementById("moose-equation-e4695acf-ee52-4c8b-b955-36b5a7ae3abc");katex.render("\\nu", element, {displayMode:false,throwOnError:false});$ (atm $var element = document.getElementById("moose-equation-2a79f2ca-657e-4c9a-bfe1-225f8567c912");katex.render("{m}^{-3} {s}^{-1}", element, {displayMode:false,throwOnError:false});$ ) in Eq. (2) and Eq. (3) represents the fact that bubbles are nucleated as dimers.

The nucleation rate is calculated as: $(4)var element = document.getElementById("moose-equation-9530c846-5efe-4c45-bc12-0f503d59c51f");katex.render(" \\nu = 8 \\pi D R_{sg}f_{n}c^{2}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-dbf8adfc-01a9-4d26-9a62-83a40ebe9c5e");katex.render("R_{sg}", element, {displayMode:false,throwOnError:false});$ (m) is the radius of a single fission gas atom and $var element = document.getElementById("moose-equation-0d4a25f1-115a-4ba7-bf10-25c1bd1ffd28");katex.render("f_{n}", element, {displayMode:false,throwOnError:false});$ is the nucleation factor (dimensionless), equal to $var element = document.getElementById("moose-equation-9108ba63-8753-496f-9c10-a6cb1276f47d");katex.render("10^{-6}", element, {displayMode:false,throwOnError:false});$ (e.g., Veshchunov (2000)). The re-solution rate is expressed as: $(5)var element = document.getElementById("moose-equation-81f61353-da37-4f27-84cf-b050fce68935");katex.render(" b = b_0 \\dot{F}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-4d489477-f1eb-46c6-ba5f-04b725859104");katex.render("\\dot{F}", element, {displayMode:false,throwOnError:false});$ is the fission rate density (m $var element = document.getElementById("moose-equation-b66ace4e-6c9f-4613-bc5d-4455a8ae5a5a");katex.render("^{-3}", element, {displayMode:false,throwOnError:false});$ s $var element = document.getElementById("moose-equation-1a2a14b6-8a1e-4b30-b360-c12626976fd9");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) The trapping rate is calculated as Ham (1958): $(6)var element = document.getElementById("moose-equation-e76b7e7f-29b0-40bd-8486-cc91458934f8");katex.render(" g = 4 \\pi D R_b c", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-a92c448e-a1d6-4429-ad66-e3fd5e8730f0");katex.render("R_b", element, {displayMode:false,throwOnError:false});$ is the average bubble radius (m). The system of equations Eq. (1),Eq. (2),Eq. (3) is solved using the recently developed PolyPole-2 algorithm (Pastore et al., 2018). Note, Polypole-1 should not be used for U3Si2 due to a relatively low resolution parameter.

Intra-granular bubble growth is computed based on the evolution of the gas atom content from Eq. (3) and the rate of absorption of vacancies at the bubble. The latter is computed using a modified Speight and Beere (1975) model, as follows.

The mechanical equilibrium of an intra-granular bubble, assumed to be spherical, is governed by the Young-Laplace equation $(7)var element = document.getElementById("moose-equation-a4c30308-5cb7-45d3-9088-9c3c7a8bca12");katex.render(" p_{eq} = \\frac{2 \\gamma}{R_b} - \\sigma_h", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-c2eb6cfb-a61b-41fe-a0cb-73d7f0c29aae");katex.render("p_{eq}", element, {displayMode:false,throwOnError:false});$ (Pa) is the equilibrium pressure, $var element = document.getElementById("moose-equation-580ccb81-0320-481a-8338-820cefb51ee9");katex.render("\\gamma", element, {displayMode:false,throwOnError:false});$ (J $var element = document.getElementById("moose-equation-00be4ea9-386b-402b-a08f-43c9774d279d");katex.render("\\mathrm{m}^{-2}", element, {displayMode:false,throwOnError:false});$ ) is the U $var element = document.getElementById("moose-equation-11b3edf9-160c-4234-bd3c-b8a9e2f21891");katex.render("_3", element, {displayMode:false,throwOnError:false});$ Si $var element = document.getElementById("moose-equation-f194e45e-b44c-437b-9f02-c153f23d15b5");katex.render("_2", element, {displayMode:false,throwOnError:false});$ /gas surface energy and $var element = document.getElementById("moose-equation-a9b6b7e7-06f4-4fc4-a550-d5fc7dcacdb8");katex.render("\\sigma_h", element, {displayMode:false,throwOnError:false});$ (Pa) is the hydrostatic stress. In general, the bubbles are in a non-equilibrium state and tend to the equilibrium condition absorbing or emitting vacancies. The vacancy absorption/emission rate can be calculated starting from the approach in Speight and Beere (1975) as $(8)var element = document.getElementById("moose-equation-3695f5d5-585a-453a-89c9-cc7d6508a894");katex.render(" \\frac{\\mathrm{d}n_{iv}}{\\mathrm{d}t}= \\dfrac{2 \\pi D^{v}_{ig} \\rho}{k T \\zeta} (p -p_{eq})", element, {displayMode:true,throwOnError:false});$ being $var element = document.getElementById("moose-equation-5802e3e5-1286-4f72-9f0a-6bfe6a49df92");katex.render("n_{iv}", element, {displayMode:false,throwOnError:false});$ (dimensionless) the number of vacancies per intra-granular bubble, $var element = document.getElementById("moose-equation-099fc223-1c16-488f-8289-f7f2dc397f8f");katex.render("D^{v}_{ig}", element, {displayMode:false,throwOnError:false});$ ( $var element = document.getElementById("moose-equation-409b349c-35cf-4300-8a36-bceed3fc89d0");katex.render("\\mathrm{m}^2", element, {displayMode:false,throwOnError:false});$ $var element = document.getElementById("moose-equation-db6760ba-af99-406d-9a31-5ccbbecdb84a");katex.render("\\mathrm{s}^{-1}", element, {displayMode:false,throwOnError:false});$ ) the intra-granular vacancy diffusion coefficient, $var element = document.getElementById("moose-equation-53fa51fb-3405-4479-9b7e-3b6850a00471");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$ (m) is the radius of the equivalent Wigner-Seitz cell surrounding a bubble and influenced by the vacancy absorption/emission, $var element = document.getElementById("moose-equation-016d8ed6-33c8-41f7-8bde-fe2e054b3be9");katex.render("k", element, {displayMode:false,throwOnError:false});$ (J K $var element = document.getElementById("moose-equation-fce16c55-796a-496a-82d0-2b2af8d053ea");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) is the Boltzmann constant, $var element = document.getElementById("moose-equation-36fa4207-c2fa-4995-ba9a-17af006aa674");katex.render("T", element, {displayMode:false,throwOnError:false});$ (K) is the local temperature, and $var element = document.getElementById("moose-equation-15690942-27a3-4736-8472-48753ae9f1b7");katex.render("\\zeta", element, {displayMode:false,throwOnError:false});$ (dimensionless) is an dimensionless factor, which is calculated as (Pizzocri et al., 2016) $(9)var element = document.getElementById("moose-equation-4f4caf23-60e1-4039-a2f2-050133f74f1c");katex.render(" \\zeta = \\dfrac{10 \\psi (1+\\psi^3)}{-\\psi^6+5\\psi^2-9\\psi+5}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-a12ff281-921f-4801-a340-c7f4bd52e0cd");katex.render("\\psi=\\dfrac{R_b}{\\rho}", element, {displayMode:false,throwOnError:false});$ is the ratio between the bubble and the cell radii. The present model for vacancy absorption/emission at intra-granular bubbles is a reformulation of the Speight and Beere model for behavior at grain boundaries of bubbles of circular projection (2D problem) (Speight and Beere, 1975). In particular, Eq. (8),Eq. (9) represent the equivalent model for vacancy absorption/emission at spherical bubbles in the bulk (3D problem). Considering a van der Waals gas, the pressure of the bubble is expressed as $(10)var element = document.getElementById("moose-equation-872f4572-c86e-48b3-94f6-14a160c381fd");katex.render(" p=\\dfrac{kT}{\\Omega-\\eta \\omega}\\eta", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-a2e901ad-4703-4cd0-80a9-43369baba849");katex.render("\\Omega", element, {displayMode:false,throwOnError:false});$ ( $var element = document.getElementById("moose-equation-9d980b3f-455d-447b-ad3e-b9e08f8baa59");katex.render("\\mathrm{m}^3", element, {displayMode:false,throwOnError:false});$ ) is the vacancy volume, $var element = document.getElementById("moose-equation-0a955c52-8001-42dc-b614-c172cdcd22ab");katex.render("\\eta", element, {displayMode:false,throwOnError:false});$ (dimensionless) is the ratio between $var element = document.getElementById("moose-equation-122df307-7602-4b85-896d-8652c9bd705a");katex.render("\\bar{n}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-fdb5cb17-9ede-4189-9c62-b6192e21e84f");katex.render("n_{iv}", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-e6456f18-1afc-4e2a-b4e0-a0c9ea142535");katex.render("\\omega", element, {displayMode:false,throwOnError:false});$ ( $var element = document.getElementById("moose-equation-523f7e6a-c9f0-48d9-9337-549f4ba85248");katex.render("\\mathrm{m}^3", element, {displayMode:false,throwOnError:false});$ ) is the van der Waals atomic volume for xenon.

The volume of intra-granular bubbles is calculated as $var element = document.getElementById("moose-equation-84f14169-3ee9-4e6e-a800-54868e702f82");katex.render("V_b=n_{iv}\\Omega", element, {displayMode:true,throwOnError:false});$

The fractional volume intra-granular fission gas swelling is given by $(11)var element = document.getElementById("moose-equation-26aa91c0-cb75-4d74-9e03-4e5da45dec21");katex.render(" \\left(\\frac{\\Delta V}{V}\\right)_{ig} = V_b N", element, {displayMode:true,throwOnError:false});$

Grain-boundary gas behavior

The numerical solution of Eq. (2) allows calculation of the arrival rate of gas at the grain boundaries, providing the source term for the grain-boundary gas behavior module. This computes both grain-boundary fission gas swelling and fission gas release through a direct description of the grain-face bubble development, including bubble growth, coalescence, and eventual inter-connection leading to fission gas release. The concept of the grain-boundary model is identical to that of the UO $var element = document.getElementById("moose-equation-d0545c3d-cc1b-4623-bb5a-de8b23c37def");katex.render("_2", element, {displayMode:false,throwOnError:false});$ ( UO2Sifgrs) model. However, the material parameters are specific to the U3Si2Sifgrs model.

The fractional volume grain-boundary fission gas swelling is given by $(12)var element = document.getElementById("moose-equation-0de2cfd1-46d9-41aa-894e-9d3210dfc6c3");katex.render(" \\left(\\frac{\\Delta V}{V}\\right)_{gf} = \\frac{1}{2} \\frac{N_{gf}}{(1/3)r_{gr}} V_{gf}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-c33f3d3f-de01-443d-abbc-e1ca8ef71539");katex.render("N_{gf}", element, {displayMode:false,throwOnError:false});$ is the number density of grain-face bubbles per unit surface, $var element = document.getElementById("moose-equation-a3185eca-2d9c-41fa-856b-a3f1e58839e1");katex.render("r_{gr}", element, {displayMode:false,throwOnError:false});$ the grain radius, and $var element = document.getElementById("moose-equation-eeb8a6bf-a743-41ef-8a71-c26bb1448d7f");katex.render("V_{gf}", element, {displayMode:false,throwOnError:false});$ the bubble volume. The factor 1/2 is introduced in Eq. (12) because a grain-face bubble is shared by two neighboring grains.

Bubble growth is calculated with the model from Speight and Beere (1975) to describe the growth (or shrinkage) of grain-face bubbles as proceeding by absorption (or emission) of vacancies in grain boundaries, induced by the difference between the pressure of the gas in the bubble, $var element = document.getElementById("moose-equation-b3606f3a-da61-40e3-bdf0-be2eb6c752b9");katex.render("p", element, {displayMode:false,throwOnError:false});$ (Pa), and the mechanical equilibrium pressure, $var element = document.getElementById("moose-equation-9f3910df-298d-42b7-9f42-d3a85ede3fc2");katex.render("p_{eq}", element, {displayMode:false,throwOnError:false});$ (Pa). The approach is conceptually analogous to that applied in for the growth and shrinkage of intra-granular bubbles and is described in UO2Sifgrs. The grain-boundary diffusion coefficient of vacancies is estimated by multiplying the intra-granular diffusion coefficient of vacancies by a factor of $var element = document.getElementById("moose-equation-7098e736-db5b-4d5e-8962-271b6dc1f280");katex.render("10^{6}", element, {displayMode:false,throwOnError:false});$ . Ultimately, the volume of a bubble comprising $var element = document.getElementById("moose-equation-e92b4445-d769-4ca7-ba6c-d2249b472f3b");katex.render("n_g", element, {displayMode:false,throwOnError:false});$ fission gas atoms and $var element = document.getElementById("moose-equation-cd423d5c-a924-48ef-abd5-2cb0aef9e1e3");katex.render("n_v", element, {displayMode:false,throwOnError:false});$ vacancies is given by $(13)var element = document.getElementById("moose-equation-c620929e-9c14-4cc7-a7fa-f667c28d8bb8");katex.render(" V_{gf} = n_g \\omega + n_v \\Omega_{gf}", element, {displayMode:true,throwOnError:false});$

An initial concentration of grain-face bubbles equal to $var element = document.getElementById("moose-equation-6ac11c94-952e-42ad-b2f1-4dfaa25b003d");katex.render("2 \\cdot 10^{12}", element, {displayMode:false,throwOnError:false});$ is employed. This value is one order of magnitude lower than the one employed in the UO $var element = document.getElementById("moose-equation-02ed6858-4c57-4fed-a601-4640faf56ef3");katex.render("_2", element, {displayMode:false,throwOnError:false});$ model, and is based on the lower bubble density noticeable in uranium silicide from the available experimental data (Shimizu, 1965). The process of grain-face bubble coalescence, which leads to a progressive decrease of the bubble number density throughout irradiation, is described with a model based on Pastore et al. (2013) and White (2004). According to this model, the evolution of the number density of grain-boundary bubbles is given by $var element = document.getElementById("moose-equation-73e9bba2-cde9-4d63-acd7-d507674e4f24");katex.render("\\frac{dN_{gf}}{dt} = -\\frac{6N_{gf}^2}{3+4N_{gf}A_{gf}}\\frac{dA_{gf}}{dt}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-29263264-ae09-427f-9111-eaf49624e19a");katex.render("N_{gf}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-64033d7b-ddb6-4325-be8f-01d71e978639");katex.render("A_{gf}", element, {displayMode:false,throwOnError:false});$ represent the number density and projected area of grain-face bubbles, respectively.

The release of fission gas to the fuel rod free volume after the inter-connection of grain-face bubbles and the consequent formation of pathways for gas venting to the fuel exterior (thermal release) is based on the principle of grain face saturation. More precisely, after the fractional coverage, $var element = document.getElementById("moose-equation-66b2d7d7-7185-40ce-9d98-aa7b20d2fc2c");katex.render("F", element, {displayMode:false,throwOnError:false});$ , attains a saturation value, $var element = document.getElementById("moose-equation-ef3cb708-1d29-41b1-bf45-6d7a83550d35");katex.render("F_{sat}", element, {displayMode:false,throwOnError:false});$ , further bubble growth is compensated by gas release in order to maintain the constant coverage condition: $(14)var element = document.getElementById("moose-equation-16991bb5-64aa-4e9a-adf0-d058896ec9b3");katex.render(" F = N_{gf}A_{gf} = F_{sat}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-6f445b2f-5f0b-44b4-9edf-d4b06b7bf9ec");katex.render("N_{gf}", element, {displayMode:false,throwOnError:false});$ is the bubble number density and $var element = document.getElementById("moose-equation-092df23e-ce28-4532-9332-8142e3cd002c");katex.render("A_{gf} = \\pi\\left(\\sin\\left(\\Theta\\right)\\right)^2R_{gf}^2", element, {displayMode:false,throwOnError:false});$ is the bubble projected area on the grain face. In the absence of experimental data or calculations on the maximum grain-face bubble coverage in U $var element = document.getElementById("moose-equation-b5452cdf-3362-4e3a-9a9c-25db8c1ff0d0");katex.render("_3", element, {displayMode:false,throwOnError:false});$ Si $var element = document.getElementById("moose-equation-3eea3ca9-d65f-4fca-aaf0-287c7b223625");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , the same value used for UO $var element = document.getElementById("moose-equation-177e614e-4ba1-4254-b792-08a49adef147");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , i.e., $var element = document.getElementById("moose-equation-61944cdd-3d3f-4e26-827c-9eab91576e60");katex.render("F_{sat}=0.5", element, {displayMode:false,throwOnError:false});$ , is considered.

Example Input Syntax

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [fission_gas_behavior]
    type = U3Si2Sifgrs<<<{"description": "Computes fission gas release and swelling in U3Si2 through a physically-based model.", "href": "U3Si2Sifgrs.html"}>>>
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 0
    temperature<<<{"description": "Coupled temperature"}>>> = T
    fission_rate<<<{"description": "Coupled fission rate variable (fiss/m^3/s)"}>>> = fission_rate
    grain_radius_const<<<{"description": "Constant grain radius (m)"}>>> = 28.e-06
  []
[]

Input Parameters

temperatureCoupled temperature
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled temperature

Required Parameters

atomic_covolume8.47705e-29Van der Waals covolume for Xe (m^3/atm)
Default:8.47705e-29
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Van der Waals covolume for Xe (m^3/atm)
blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
boundaryThe list of boundaries (ids or names) from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundaries (ids or names) from the mesh where this object applies
bubble_gb_limit1e+10grain-boundary bubble number density limit (bbl/m**2)
Default:1e+10
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:grain-boundary bubble number density limit (bbl/m**2)
burnupCoupled burnup
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled burnup
burnup_functionBurnup function
C++ Type:BurnupFunctionName
Unit:(no unit assumed)
Controllable:No
Description:Burnup function
computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
Default:True
C++ Type:bool
Controllable:No
Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
Default:NONE
C++ Type:MooseEnum
Options:NONE, ELEMENT, SUBDOMAIN
Controllable:No
Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
declare_suffixAn optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
diff_coeff_optionSI_VACANCY_STOICHIOMETRYSelect diffusion coefficient
Default:SI_VACANCY_STOICHIOMETRY
C++ Type:MooseEnum
Options:U_VACANCY, U_VACANCY_ANISOTROPY, U_VACANCY_BARANI, SI_VACANCY_STOICHIOMETRY, SI_VACANCY_RICH, TEST_CASE
Controllable:No
Description:Select diffusion coefficient
dislocation_bubble_nucleation_factor1e+06dislocation bubble nucleation factor, i.e., number of bubbles per dislocation line density (bubbles/(m/m2))
Default:1e+06
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:dislocation bubble nucleation factor, i.e., number of bubbles per dislocation line density (bubbles/(m/m2))
dislocation_core_radius3.85e-10dislocation core radius (m), i.e., burgers vector length
Default:3.85e-10
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:dislocation core radius (m), i.e., burgers vector length
dislocation_density40000000000000.0Coupled dislocation density (m/m^3)
Default:40000000000000.0
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled dislocation density (m/m^3)
dislocation_density_materialdislocation density material property (m/m^3)
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:dislocation density material property (m/m^3)
dislocation_punchingFalseFlag to allow dislocation punching, effectively limiting bubble pressures and tracking the rate of dislocation production.
Default:False
C++ Type:bool
Controllable:No
Description:Flag to allow dislocation punching, effectively limiting bubble pressures and tracking the rate of dislocation production.
dislocation_trap_Zfactor25Z * dislocation core radius = dislocation_trap_radius (m)
Default:25
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Z * dislocation core radius = dislocation_trap_radius (m)
effdiffcoeff_scalef1Scaling factor for intragranular effective diffusion coefficient
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scaling factor for intragranular effective diffusion coefficient
eos_optionORIGINAL_MODELSelect Equation of State
Default:ORIGINAL_MODEL
C++ Type:MooseEnum
Options:ORIGINAL_MODEL, YANG
Controllable:No
Description:Select Equation of State
fission_gas_concCoupled fission gas concentration
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled fission gas concentration
fission_rateCoupled fission rate variable (fiss/m^3/s)
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled fission rate variable (fiss/m^3/s)
fission_rate_materialFission rate material property (fiss/m^3/s)
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Fission rate material property (fiss/m^3/s)
fract_yield0.3Fractional yield of fission gas atoms (Xe + Kr) (/)
Default:0.3
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Fractional yield of fission gas atoms (Xe + Kr) (/)
gas_atom_dislocation_diffusion_multiplier1Factor by which gas atom diffusion is multiplied along dislocations compared to the bulk to capture pipe diffusion
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Factor by which gas atom diffusion is multiplied along dislocations compared to the bulk to capture pipe diffusion
gas_diffusivity_functionOptional function for gas atomic diffusivity (m^2/s). For implementation of the function, the x-axis corresponds to temperature and y-axis corresponds to fission rate. The z-axis is unused, and time is the standard current time.
C++ Type:FunctionName
Unit:(no unit assumed)
Controllable:No
Description:Optional function for gas atomic diffusivity (m^2/s). For implementation of the function, the x-axis corresponds to temperature and y-axis corresponds to fission rate. The z-axis is unused, and time is the standard current time.
gbdiffcoeff_scalef1Scaling factor for grain-boundary vacancy diffusion coefficient
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scaling factor for grain-boundary vacancy diffusion coefficient
grain_radiusCoupled grain Radius
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled grain Radius
grain_radius_const2.5e-05Constant grain radius (m)
Default:2.5e-05
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Constant grain radius (m)
grainradius_scalef1Scaling factor for grain radius
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scaling factor for grain radius
hydrostatic_stressCoupled hydrostatic Stress
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled hydrostatic Stress
hydrostatic_stress_const0constant hydrostatic stress (Pa)
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:constant hydrostatic stress (Pa)
ig_bubble_coarseningNO_COARSENINGSelect intra-granular diffusion algorithm
Default:NO_COARSENING
C++ Type:MooseEnum
Options:NO_COARSENING, WITH_COARSENING
Controllable:No
Description:Select intra-granular diffusion algorithm
ig_bubble_modelNUCLEATION_RESOLUTIONSelect bubble evolution model
Default:NUCLEATION_RESOLUTION
C++ Type:MooseEnum
Options:NUCLEATION_RESOLUTION
Controllable:No
Description:Select bubble evolution model
ig_diff_algorithmPOLYPOLE2Select intra-granular diffusion algorithm
Default:POLYPOLE2
C++ Type:MooseEnum
Options:POLYPOLE2
Controllable:No
Description:Select intra-granular diffusion algorithm
ig_fully_coupledFULLY_COUPLEDSolving diffusion coupled to bubble evolution
Default:FULLY_COUPLED
C++ Type:MooseEnum
Options:FULLY_COUPLED
Controllable:No
Description:Solving diffusion coupled to bubble evolution
igdiffcoeff_scalef1Scaling factor for intragranular diffusion coefficients
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scaling factor for intragranular diffusion coefficients
initial_porosity0Initial fuel porosity (/)
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Initial fuel porosity (/)
intergranular_bubble_coalescence_optionWHITE2004Select intergranular bubble coalescence model
Default:WHITE2004
C++ Type:MooseEnum
Options:WHITE2004, PASTORE2013
Controllable:No
Description:Select intergranular bubble coalescence model
minimum_pressure_ratio1Factor to limit the equilibrium pressure of bubbles by enforcing a minimum ratio of the equilibrium pressure to the pressure due to surface energies alone. If greater than one, tensile stresses do not impact the equilibrium bubble size. This factor is not allowed to be below 0 since it can result in negative equilibrium pressures. A value of of 0.1 - 0.5 is reasonable for capturing tensile stress impacts on the equilibrium bubble pressure.
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Factor to limit the equilibrium pressure of bubbles by enforcing a minimum ratio of the equilibrium pressure to the pressure due to surface energies alone. If greater than one, tensile stresses do not impact the equilibrium bubble size. This factor is not allowed to be below 0 since it can result in negative equilibrium pressures. A value of of 0.1 - 0.5 is reasonable for capturing tensile stress impacts on the equilibrium bubble pressure.
nucleation_optionHOMOGENEOUSSelect intragranular bubble nucleation model
Default:HOMOGENEOUS
C++ Type:MooseEnum
Options:HOMOGENEOUS
Controllable:No
Description:Select intragranular bubble nucleation model
nuclerate_scalef1Scaling factor for nucleation rate
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scaling factor for nucleation rate
percolation_to_surface1.0Optional AuxVariable that indicates whether the local position is connected by a percolated path to a free surface to allow gas release. If this parameter is not set, path to free surface is not considered in the gas release calculation.
Default:1.0
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Optional AuxVariable that indicates whether the local position is connected by a percolated path to a free surface to allow gas release. If this parameter is not set, path to free surface is not considered in the gas release calculation.
res_param_optionHOMOGENEOUS_MATTHEWSSelect resolution parameter
Default:HOMOGENEOUS_MATTHEWS
C++ Type:MooseEnum
Options:HOMOGENEOUS_MATTHEWS, HETEROGENEOUS_MATTHEWS, TEST_CASE
Controllable:No
Description:Select resolution parameter
resolutionp_scalef1Scaling factor for resolution parameter
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scaling factor for resolution parameter
resolutionp_scalef_dislocation1Scaling factor for resolution parameter in dislocations
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scaling factor for resolution parameter in dislocations
saturation_coverage0.5initial grain boundary saturation coverage (/)
Default:0.5
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:initial grain boundary saturation coverage (/)
semidihedral_angle1.27235Semi-dihedral angle of grain-boundary bubbles (rad)
Default:1.27235
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Semi-dihedral angle of grain-boundary bubbles (rad)
shear_modulusShear modulus material property (Pa). Required if dislocation_punching=true
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Shear modulus material property (Pa). Required if dislocation_punching=true
surface_energy1.7U3Si2-gas surface energy (J/m^2)
Default:1.7
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:U3Si2-gas surface energy (J/m^2)
temperature_scalef1Scaling factor for temperature
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scaling factor for temperature
trap_param_optionDEFAULTSelect trapping parameter
Default:DEFAULT
C++ Type:MooseEnum
Options:DEFAULT, TEST_CASE
Controllable:No
Description:Select trapping parameter
trappingp_scalef1Scaling factor for trapping parameter
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scaling factor for trapping parameter
trappingp_scalef_dislocation1Scaling factor for trapping parameter in dislocation bubbles
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scaling factor for trapping parameter in dislocation bubbles
trappingp_scalef_dislocation_line1Scaling factor for trapping parameter in dislocation lines
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scaling factor for trapping parameter in dislocation lines
vacancies_per_atom_diffusion_functionOptional function for the number of vacancies assisting gas atom diffusion.
C++ Type:FunctionName
Unit:(no unit assumed)
Controllable:No
Description:Optional function for the number of vacancies assisting gas atom diffusion.
vacancy_diffusivity_functionOptional function for vacancy diffusivity (m^2/s). For implementation of the function, the x-axis corresponds to temperature and y-axis corresponds to fission rate. The z-axis is unused, and time is the standard current time.
C++ Type:FunctionName
Unit:(no unit assumed)
Controllable:No
Description:Optional function for vacancy diffusivity (m^2/s). For implementation of the function, the x-axis corresponds to temperature and y-axis corresponds to fission rate. The z-axis is unused, and time is the standard current time.
vacancy_dislocation_diffusion_multiplier1Factor by which vacancy diffusion is multiplied along dislocations compared to the bulk to capture pipe diffusion
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Factor by which vacancy diffusion is multiplied along dislocations compared to the bulk to capture pipe diffusion
vacancy_volume4.09e-29Atomic (vacancy) volume in bubbles (m^3)
Default:4.09e-29
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Atomic (vacancy) volume in bubbles (m^3)

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator
skip_bdr_modelFalseSkips the grain-boundary model
Default:False
C++ Type:bool
Controllable:No
Description:Skips the grain-boundary model
testing_outputFalseProvides an analytic reference for the value of the intra-granular fission gas release
Default:False
C++ Type:bool
Controllable:No
Description:Provides an analytic reference for the value of the intra-granular fission gas release
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)
C++ Type:std::vector<std::string>
Controllable:No
Description:List of material properties, from this material, to output (outputs must also be defined to an output type)
outputsnone Vector of output names where you would like to restrict the output of variables(s) associated with this object
Default:none
C++ Type:std::vector<OutputName>
Controllable:No
Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

Input Files

(test/tests/sifgrs/u3si2/intragranular.i)
(test/tests/sifgrs/u3si2/negative_temperature_exception.i)
(test/tests/sifgrs/u3si2/intergranular_ext_fsngas.i)
(examples/accident_tolerant_fuel/u3si2_zircaloy/u3si2_zircaloy.i)
(test/tests/sifgrs/u3si2/intergranular.i)
(test/tests/sifgrs/u3si2/polypole2.i)
(test/tests/sifgrs/u3si2/option_base.i)
(examples/accident_tolerant_fuel/u3si2_sic/u3si2_outer_monolith_1.5D.i)
(test/tests/sifgrs/u3si2/burnup_function.i)
(test/tests/sifgrs/u3si2/polypole2_ext_fsngas.i)
(test/tests/solid_mechanics/u3si2_eigenstrains/u3si2_vswelling/swelling_mechanistic.i)

References

D.A. Andersson. Density functional theory calculations of the defect and fission gas properties in U-Si fuels. Technical Report, Los Alamos National Laboratory, 2017.

@techreport{andersson_2017,
    author = "Andersson, D.A.",
    title = "Density functional theory calculations of the defect and fission gas properties in {U-Si} fuels",
    year = "2017",
    institution = "Los Alamos National Laboratory"
}

B. Beeler, M. Baskes, D. Andersson, M. W. D. Cooper, and Y. Zhang. Molecular dynamics investigation of grain boundaries and surfaces in u$_3$si$_2$. Journal of Nuclear Materials, 514:290–298, 2019.

@ARTICLE{beeler_et_al_2019,
    author = "Beeler, B. and Baskes, M. and Andersson, D. and Cooper, M. W. D. and Zhang, Y.",
    title = "Molecular dynamics investigation of grain boundaries and surfaces in U$\_3$Si$\_2$",
    journal = "Journal of Nuclear Materials",
    year = "2019",
    volume = "514",
    pages = "290-298"
}

F.S. Ham. Theory of diffusion-limited precipitation. J. Phys. Chem. Solids, 6:335–351, 1958.

@ARTICLE{ham_1958,
    author = "Ham, F.S.",
    title = "Theory of diffusion-limited precipitation",
    journal = "J. Phys. Chem. Solids",
    year = "1958",
    volume = "6",
    pages = "335-351"
}

C. Matthews, D.A. Andersson, and C. Unal. Radiation re-solution calculation in uranium-silicide fuels. Technical Report, Los Alamos National Laboratory, 2016.

@techreport{matthews_et_al_2016,
    author = "Matthews, C. and Andersson, D.A. and Unal, C.",
    title = "Radiation re-solution calculation in uranium-silicide fuels",
    year = "2016",
    institution = "Los Alamos National Laboratory"
}

D. R. Olander and D. Wongsawaeng. Re-solution of fission gas - A review: Part I. Intragranular bubbles. Journal of Nuclear Materials, 354:94–109, 2006.

@article{olander_and_wongsawaeng_2006,
    author = "Olander, D. R. and Wongsawaeng, D.",
    title = "Re-solution of fission gas - {A} review: {P}art {I}. {I}ntragranular bubbles",
    journal = "Journal of Nuclear Materials",
    year = "2006",
    volume = "354",
    pages = "94-109"
}

G. Pastore, L. Luzzi, V. Di Marcello, and P. Van Uffelen. Physics-based modelling of fission gas swelling and release in UO$_2$ applied to integral fuel rod analysis. Nuclear Engineering and Design, 256:75–86, 2013.

@ARTICLE{pastore_et_al_2013,
    author = "Pastore, G. and Luzzi, L. and Marcello, V. Di and Uffelen, P. Van",
    title = "Physics-based modelling of fission gas swelling and release in {UO}$\_2$ applied to integral fuel rod analysis",
    journal = "Nuclear Engineering and Design",
    year = "2013",
    volume = "256",
    pages = "75-86"
}

G. Pastore, D. Pizzocri, C. Rabiti, T. Barani, P. Van Uffelen, and L.. Luzzi. An effective numerical algorithm for intra-granular fission gas release during non-equilibrium trapping and resolution. Journal of Nuclear Materials, 509:687–699, 2018.

@ARTICLE{pastore_et_al_2018,
    author = "Pastore, G. and Pizzocri, D. and Rabiti, C. and Barani, T. and Van Uffelen, P. and Luzzi, L..",
    title = "An effective numerical algorithm for intra-granular fission gas release during non-equilibrium trapping and resolution",
    journal = "Journal of Nuclear Materials",
    year = "2018",
    volume = "509",
    pages = "687-699"
}

D. Pizzocri, F. Cappia, V. V. Rondinella, and P. Van Uffelen. Preliminary model for the pore growth in the HBS. Technical Report, JRC103064, European Commission, Directorate for Nuclear Safety and Security, JRC-Karlsruhe, 2016.

@techreport{pizzocri_jrc_rep_2016,
    author = "Pizzocri, D. and Cappia, F. and Rondinella, V. V. and Van Uffelen, P.",
    title = "Preliminary model for the pore growth in the {HBS}",
    year = "2016",
    institution = "JRC103064, European Commission, Directorate for Nuclear Safety and Security, JRC-Karlsruhe"
}

H. Shimizu. The properties and irradiation behavior of U$_3$Si$_2$. Technical Report NAA-SR-10621, Atomics International, 1965.

@TECHREPORT{Shimizu_1965,
    author = "Shimizu, H.",
    title = "{The properties and irradiation behavior of U$\_3$Si$\_2$}",
    institution = "Atomics International",
    year = "1965",
    number = "NAA-SR-10621"
}

M.V. Speight and W. Beere. Vacancy potential and void growth on grain boundaries. Metal Science, 9:190–191, 1975.

@ARTICLE{speightandbeere1975,
    author = "Speight, M.V. and Beere, W.",
    title = "Vacancy Potential and Void Growth on Grain Boundaries",
    journal = "Metal Science",
    year = "1975",
    volume = "9",
    pages = "190-191"
}

M. S. Veshchunov. On the theory of fission gas bubble evolution in irradiated UO$_2$ fuel. Journal of Nuclear Materials, 277:67–81, 2000.

@article{Veshchunov2000,
    author = "Veshchunov, M. S.",
    journal = "Journal of Nuclear Materials",
    title = "{On the theory of fission gas bubble evolution in irradiated {UO}$\_2$ fuel}",
    volume = "277",
    year = "2000",
    pages = "67-81"
}

R.J. White. The development of grain-face porosity in irradiated oxide fuel. Journal of Nuclear Materials, 325:61–77, 2004.

@ARTICLE{white2004,
    author = "White, R.J.",
    title = "The development of grain-face porosity in irradiated oxide fuel",
    journal = "Journal of Nuclear Materials",
    year = "2004",
    volume = "325",
    pages = "61-77"
}

(test/tests/solid_mechanics/u3si2_eigenstrains/u3si2_vswelling/swelling_mechanistic.i)

# This test verifies the overall U3Si2 fgb model through a single element calculation.
# The outcome has been benchmarked against the result of a stand-alone, independent software.
# In this input file, the coupled fgr and gaseous swelling calculation is performed.

initial_fuel_density = 11590.0

[GlobalParams]
  density = ${initial_fuel_density}
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 3
    xmin = 0
    xmax = 0.01
    ymin = 0
    ymax = 0.01
    zmin = 0
    zmax = 0.01
    nx = 1
    ny = 1
    nz = 1
  []
[]

[Variables]
  [T]
    order = FIRST
    family = LAGRANGE
    initial_condition = 1250.0
  []
[]

[AuxVariables]
  [fission_rate]
    order = FIRST
    family = LAGRANGE
  []
  [burnup]
    order = CONSTANT
    family = MONOMIAL
  []
  [rad_bbl_grn]
    order = CONSTANT
    family = MONOMIAL
  []
  [bbl_grn_3]
    order = CONSTANT
    family = MONOMIAL
  []
  [gas_bbl_grn]
    order = CONSTANT
    family = MONOMIAL
  []
  [gas_bdr_3]
    order = CONSTANT
    family = MONOMIAL
  []
  [deltav_v0_intra_total]
    order = CONSTANT
    family = MONOMIAL
  []
  [deltav_v0_bubble_GB]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = T
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = FINITE
    add_variables = true
    eigenstrain_names = 'fuelthermal_strain volumetric_swelling'
    temperature = T
  []
[]

[Functions]
  [Temp_func]
    type = ParsedFunction
    expression = '1250'
  []
  [Fiss_func]
    type = ParsedFunction
    expression = '2.e19'
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable = fission_rate
    value = 1
    fission_rate_function = Fiss_func
    execute_on = 'initial  timestep_begin'
  []
  [burnup]
    type = BurnupAux
    variable = burnup
    fission_rate = fission_rate
  []
  [bbl_cnc]
    type = MaterialRealAux
    variable = bbl_grn_3
    property = bubble_concentration_intra
  []
  [rad_bbl]
    type = MaterialRealAux
    variable = rad_bbl_grn
    property = bubble_radius_intra
  []
  [gascnc_bbl]
    type = MaterialRealAux
    variable = gas_bbl_grn
    property = gas_concentration_bubble_intra
  []
  [dvv0gr]
    type = MaterialRealAux
    variable = deltav_v0_intra_total
    property = deltav_v0_intra_total
  []
  [dvv0bd]
    type = MaterialRealAux
    variable = deltav_v0_bubble_GB
    property = deltav_v0_bubble_GB
  []
[]

[BCs]
  [bottom_y]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0.0
  []
  [left_x]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  []
  [back_z]
    type = DirichletBC
    variable = disp_z
    boundary = back
    value = 0.0
  []
  [bottom_T]
    type = FunctionDirichletBC
    variable = T
    function = Temp_func
    boundary = bottom
  []
[]

[Materials]
  [fuel_elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    block = 0
    youngs_modulus = 1e11
    poissons_ratio = 0.17
  []
  [fission_gas_behavior]
    type = U3Si2Sifgrs
    block = 0
    temperature = T
    fission_rate = fission_rate
    grain_radius_const = 28.e-06
  []
  [fuel_swelling]
    type = U3Si2VolumetricSwellingEigenstrain
    temperature = T
    burnup = burnup
    complete_burnup = 10
    eigenstrain_name = volumetric_swelling
    gaseous_swelling_type = U3SI2FG
  []
  [fuel_thermal_strain]
    type = ComputeThermalExpansionEigenstrain
    block = 0
    thermal_expansion_coeff = 1.5e-5
    temperature = T
    stress_free_temperature = 1250.0
    eigenstrain_name = fuelthermal_strain
  []
  [fuel_stress]
    type = ComputeFiniteStrainElasticStress
    block = 0
  []
  [U3Si2_thermal]
    type = HeatConductionMaterial
    block = 0
    thermal_conductivity = 1.0
    specific_heat = 1.0
  []
  [density]
    type = StrainAdjustedDensity
    block = 0
    strain_free_density = ${initial_fuel_density}
  []
[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'

  petsc_options = '-snes_ksp_ew '
  petsc_options_iname = '-ksp_gmres_restart -pc_type -pc_hypre_type'
  petsc_options_value = '70 hypre boomeramg'
  l_max_its = 60
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-15
  l_tol = 1e-5

  start_time = 0.0
  dt = 100.
  num_steps = 10
[]

[Postprocessors]
  [fis_gas_generated]
    type = ElementIntegralFisGasGeneratedSifgrs
    block = 0
  []
  [fis_gas_released]
    type = ElementIntegralFisGasReleasedSifgrs
    block = 0
  []
  [gbswelling]
    type = ElementAverageValue
    variable = deltav_v0_bubble_GB
    block = 0
  []
  [igswelling]
    type = ElementAverageValue
    variable = deltav_v0_intra_total
    block = 0
  []
[]

[Outputs]
  exodus = true
[]

(test/tests/sifgrs/u3si2/intragranular.i)

# @Requirement F2.40
# This test is for evaluating the extended PolyPole-2 algorithm (for the intra-granular behavior calculation)
# in the U3Si2Sifgrs fission gas behavior model.
#
[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 3
    xmin = 0
    xmax = 0.01
    ymin = 0
    ymax = 0.01
    zmin = 0
    zmax = 0.01
    nx = 1
    ny = 1
    nz = 1
  []
[]

[Functions]
  [Temp_func]
    type = ParsedFunction
    expression = '800'
  []
  [Fiss_func]
    type = ParsedFunction
    expression = '1.e19'
  []
[]

[Variables]
  [T]
    order = FIRST
    family = LAGRANGE
    initial_condition = 800
  []
[]

[AuxVariables]
  [fission_rate]
    order = FIRST
    family = LAGRANGE
  []
  [rad_bbl_grn]
    order = CONSTANT
    family = MONOMIAL
  []
  [bbl_grn_3]
    order = CONSTANT
    family = MONOMIAL
  []
  [gas_bbl_grn]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
    diffusion_coefficient = 1
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable = fission_rate
    value = 1
    fission_rate_function = Fiss_func
    execute_on = 'initial timestep_begin'
  []
  [bbl_cnc]
    type = MaterialRealAux
    variable = bbl_grn_3
    property = bubble_concentration_intra
  []
  [rad_bbl]
    type = MaterialRealAux
    variable = rad_bbl_grn
    property = bubble_radius_intra
  []
  [gascnc_bbl]
    type = MaterialRealAux
    variable = gas_bbl_grn
    property = gas_concentration_bubble_intra
  []
[]

[BCs]
  [bottom_T]
    type = FunctionDirichletBC
    variable = T
    function = Temp_func
    boundary = bottom
  []
[]

[Materials]
  [fission_gas_behavior]
    type = U3Si2Sifgrs
    block = 0
    skip_bdr_model = true
    temperature = T
    fission_rate = fission_rate
    grain_radius_const = 28.e-6
  []
[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'

  l_tol = 1e-4
  nl_abs_tol = 1e-5
  nl_rel_tol = 1e-5
  start_time = 0.0
  num_steps = 50
  dt = 1e3
[]

[Postprocessors]
  [fis_gas_generated]
    type = ElementIntegralFisGasGeneratedSifgrs
    block = 0
  []
  [fis_gas_released]
    type = ElementIntegralFisGasReleasedSifgrs
    block = 0
  []
[]

[Outputs]
  exodus = true
[]

(test/tests/sifgrs/u3si2/negative_temperature_exception.i)

# This test ensures that the moose exception to cut the timestep when a
# negative temperature is detected in U3Si2Sifgrs is properly executed.

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    dim = 2
  []
[]

[Functions]
  [temperature_func]
    type = PiecewiseLinear
    x = '0 2'
    y = '300 -100.0'
  []
  [Fiss_func]
    type = ConstantFunction
    value = 1e19
  []
[]

[Variables]
  [T]
    order = FIRST
    family = LAGRANGE
    initial_condition = 800.0
  []
[]

[AuxVariables]
  [fission_rate]
    order = FIRST
    family = LAGRANGE
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
    diffusion_coefficient = 1
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable = fission_rate
    value = 1
    fission_rate_function = Fiss_func
    execute_on = 'initial timestep_begin'
  []
[]

[BCs]
  [all_T]
    type = FunctionDirichletBC
    variable = T
    function = temperature_func
    boundary = 'bottom right left top'
  []
[]

[Materials]
  [fission_gas_release]
    type = U3Si2Sifgrs
    temperature = T
    fission_rate = fission_rate
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  start_time = 0.0
  num_steps = 2
  dt = 2
[]

[Postprocessors]
  [dt]
    type = TimestepSize
  []
[]

[UserObjects]
  [terminator]
    type = Terminator
    expression = 'dt = 1'
    execute_on = timestep_end
  []
[]

[Outputs]
  csv = true
[]

(test/tests/sifgrs/u3si2/intergranular_ext_fsngas.i)

# This input tests external fission gas coupling to U3Si2Sifgrs with PolyPole-2

[Mesh]
  [mesh]
    type = FileMeshGenerator
    file = 1hex8_10mm_cube.e
  []
[]

[Functions]
  [Temp_func]
    type = ParsedFunction
    expression = '1700'
  []
  [Fiss_func]
    type = ParsedFunction
    expression = '2.5e19'
  []
[]

[Variables]
  [T]
    initial_condition = 1700
  []
[]

[AuxVariables]
  [fission_rate]
  []
  [ext_gas]
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
    diffusion_coefficient = 1
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable = fission_rate
    value = 1
    fission_rate_function = Fiss_func
    execute_on = 'initial timestep_begin'
  []
  [ext_gas_aux]
    type = ParsedAux
    variable = ext_gas
    coupled_variables = 'fission_rate'
    use_xyzt = true
    expression = 'fission_rate * 0.3017 / 6.02214076e23 * t'
  []
[]

[BCs]
  [bottom_T]
    type = FunctionDirichletBC
    variable = T
    function = Temp_func
    boundary = 1
  []
[]

[Materials]
  [fission_gas_behavior]
    type = U3Si2Sifgrs
    skip_bdr_model = true
    temperature = T
    fission_rate = fission_rate
    fission_gas_conc = ext_gas
    ig_diff_algorithm = POLYPOLE2
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  l_tol = 1e-4
  nl_abs_tol = 1e-5
  nl_rel_tol = 1e-5
  start_time = 0.0
  num_steps = 50
  dt = 1e6
[]

[Postprocessors]
  [fis_gas_generated]
    type = ElementIntegralFisGasGeneratedSifgrs
  []
  [fis_gas_released]
    type = ElementIntegralFisGasReleasedSifgrs
  []
  [fgr_percent]
    type = FGRPercent
    fission_gas_released = fis_gas_released
    fission_gas_generated = fis_gas_generated
  []
[]

[Outputs]
  exodus = true
  hide = 'ext_gas'
[]

(examples/accident_tolerant_fuel/u3si2_zircaloy/u3si2_zircaloy.i)


initial_fuel_density = 11590.0

[GlobalParams]
  # Set initial fuel density, other global parameters
  density = ${initial_fuel_density}
  order = SECOND
  family = LAGRANGE
  energy_per_fission = 3.2e-11 # J/fission
  volumetric_locking_correction = false
  displacements = 'disp_x disp_y'
[]

[Problem]
  type = ReferenceResidualProblem
  reference_vector = 'ref'
  extra_tag_vectors = 'ref'
[]

[Mesh]
  coord_type = RZ
  # Import mesh file
  patch_size = 10 # For contact algorithm
  patch_update_strategy = auto
  partitioner = centroid
  centroid_partitioner_direction = y
  [mesh]
    type = FileMeshGenerator
    file = u3si2_zircaloy_smeared.e
  []
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
  [temp]
    initial_condition = 293.0
  []
[]

[UserObjects]
  [pin_geometry]
    type = FuelPinGeometry
    clad_inner_wall = 5
    clad_outer_wall = 2
    clad_top = 3
    clad_bottom = 1
    pellet_exteriors = 8
  []
[]

[AuxVariables]
  # Define auxilary variables
  [fast_neutron_flux]
    block = clad
  []
  [fast_neutron_fluence]
    block = clad
  []
  [gap_cond]
    order = CONSTANT
    family = MONOMIAL
  []
  [hoop_stress]
    order = CONSTANT
    family = MONOMIAL
  []
  [coolant_htc]
    order = CONSTANT
    family = MONOMIAL
  []
  [oxide_thickness]
    order = CONSTANT
    family = MONOMIAL
  []
  [total_hoop_strain]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_rate]
    order = CONSTANT
    family = MONOMIAL
  []
  [densification]
    order = CONSTANT
    family = MONOMIAL
  []
  [solid_swell]
    order = CONSTANT
    family = MONOMIAL
  []
  [gaseous_swell]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [power_history]
    type = PiecewiseLinear
    x = '0 1e4 1e8'
    y = '0 2.5e4 2.5e4'
    scale_factor = 1
  []
  [axial_peaking_factors]
    type = ParsedFunction
    expression = 1
  []
  [pressure_ramp]
    type = PiecewiseLinear
    x = '-200 0 1e8'
    y = '6.537e-3 1 1'
    scale_factor = 15.5e6
  []
  [mass_flux_func]
    type = PiecewiseLinear
    x = '-200 0 1e8'
    y = '3800 3800 3800'
  []
  [q]
    type = CompositeFunction
    functions = 'power_history axial_peaking_factors'
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [pellets]
    block = pellet_type_1
    strain = FINITE
    eigenstrain_names = 'fuel_thermal_strain fuel_volumetric_strain'
    generate_output = 'vonmises_stress stress_xx stress_yy stress_zz'
    extra_vector_tags = 'ref'
  []
  [clad]
    block = clad
    strain = FINITE
    eigenstrain_names = 'clad_thermal_eigenstrain clad_irradiation_strain'
    generate_output = 'vonmises_stress stress_xx stress_yy stress_zz'
    extra_vector_tags = 'ref'
  []
[]

[Kernels]
  [gravity]
    type = Gravity
    variable = disp_y
    value = -9.81
    extra_vector_tags = 'ref'
  []
  [heat]
    type = HeatConduction
    variable = temp
    extra_vector_tags = 'ref'
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temp
    extra_vector_tags = 'ref'
  []
  [heat_source]
    type = NeutronHeatSource
    variable = temp
    block = pellet_type_1
    burnup_function = burnup
    extra_vector_tags = 'ref'
  []
[]

[Burnup]
  [burnup]
    block = pellet_type_1
    rod_ave_lin_pow = power_history
    axial_power_profile = axial_peaking_factors
    num_radial = 81
    num_axial = 11
    fuel_pin_geometry = pin_geometry
    fuel_volume_ratio = 1.0
    RPF = RPF
    fuel_type = U3Si2
  []
[]

[AuxKernels]
  [fast_neutron_flux]
    type = FastNeutronFluxAux
    variable = fast_neutron_flux
    block = clad
    rod_ave_lin_pow = power_history
    axial_power_profile = axial_peaking_factors
    factor = 3e13
    execute_on = timestep_begin
  []
  [fast_neutron_fluence]
    type = FastNeutronFluenceAux
    variable = fast_neutron_fluence
    block = clad
    fast_neutron_flux = fast_neutron_flux
    execute_on = timestep_begin
  []
  [hoop_stress]
    type = RankTwoScalarAux
    rank_two_tensor = stress
    variable = hoop_stress
    scalar_type = HoopStress
    execute_on = timestep_end
  []
  [total_hoop_strain]
    type = RankTwoScalarAux
    rank_two_tensor = total_strain
    variable = total_hoop_strain
    scalar_type = HoopStress
    execute_on = timestep_end
  []
  [conductance]
    type = MaterialRealAux
    property = gap_conductance
    variable = gap_cond
    boundary = 10
  []
  [coolant_htc]
    type = MaterialRealAux
    property = coolant_channel_htc
    variable = coolant_htc
    boundary = 2
  []
  [oxide]
    type = MaterialRealAux
    variable = oxide_thickness
    property = oxide_scale_thickness
    boundary = 2
  []
  [creep_rate]
    type = MaterialRealAux
    variable = creep_rate
    property = creep_rate
    execute_on = timestep_end
    block = clad
  []
  [densfication]
    type = MaterialRealAux
    property = densification
    variable = densification
    block = pellet_type_1
  []
  [solid_swell]
    type = MaterialRealAux
    property = solid_swelling
    variable = solid_swell
    block = pellet_type_1
  []
  [gaseous_swell]
    type = MaterialRealAux
    property = gaseous_swelling
    variable = gaseous_swell
    block = pellet_type_1
  []
[]

[Contact]
  [pellet_clad_mechanical]
    primary = 5
    secondary = 10
    normal_smoothing_distance = 0.1
    penalty = 1e7
  []
[]

[ThermalContact]
  [thermal_contact]
    type = GasGapHeatTransfer
    variable = temp
    primary = 5
    secondary = 10
    initial_moles = initial_moles
    gas_released = fis_gas_released
    contact_pressure = contact_pressure
    quadrature = true
  []
[]

[BCs]
  [no_x_all] # pin pellets and clad along axis of symmetry (y)
    type = DirichletBC
    variable = disp_x
    boundary = 12
    value = 0.0
  []
  [no_y_clad_bottom] # pin clad bottom in the axial direction (y)
    type = DirichletBC
    variable = disp_y
    boundary = '1'
    value = 0.0
  []
  [no_y_fuel_bottom] # pin fuel bottom in the axial direction (y)
    type = DirichletBC
    variable = disp_y
    boundary = '1020'
    value = 0.0
  []
  [Pressure]
    [coolantPressure]
      boundary = '1 2 3'
      function = pressure_ramp
    []
  []
  [PlenumPressure]
    [plenumPressure]
      boundary = 9
      initial_pressure = 2.0e6
      startup_time = 0
      R = 8.3143
      output_initial_moles = initial_moles
      temperature = ave_temp_interior
      volume = gas_volume
      material_input = fis_gas_released
      output = plenum_pressure
    []
  []
[]

[CoolantChannel]
  [convective_clad_surface]
    boundary = '1 2 3'
    variable = temp
    inlet_temperature = 580 # K
    inlet_pressure = pressure_ramp # Pa
    inlet_massflux = mass_flux_func # kg/m^2-sec
    rod_diameter = 9.4996e-3 # m
    rod_pitch = 1.26e-2 # m
    linear_heat_rate = power_history
    axial_power_profile = axial_peaking_factors
  []
[]

[Materials]
  [fuel_thermal]
    type = SilicideFuelThermal
    block = pellet_type_1
    thermal_conductivity_model = WHITE
    silicon_mole_fraction = 0.4
    temperature = temp
  []
  [fuel_elasticity_tensor]
    type = U3Si2ElasticityTensor
    block = pellet_type_1
  []
  [fuel_stress]
    type = ComputeMultipleInelasticStress
    block = pellet_type_1
    tangent_operator = elastic
    inelastic_models = 'fuel_creep'
  []
  [fuel_creep]
    type = U3Si2CreepUpdate
    block = pellet_type_1
    temperature = temp
  []
  [fuel_thermal_expansion]
    type = U3Si2ThermalExpansionEigenstrain
    block = pellet_type_1
    temperature = temp
    stress_free_temperature = 295.0
    eigenstrain_name = fuel_thermal_strain
  []
  [fuel_volumetric_swelling]
    type = U3Si2VolumetricSwellingEigenstrain
    block = pellet_type_1
    gaseous_swelling_type = U3SI2FG
    temperature = temp
    burnup_function = burnup
    eigenstrain_name = fuel_volumetric_strain
  []

  [clad_thermal]
    type = ZryThermal
    temperature = temp
    block = clad
  []
  [clad_elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 7.5e10
    poissons_ratio = 0.3
    block = clad
  []
  [clad_stress]
    type = ComputeMultipleInelasticStress
    tangent_operator = elastic
    inelastic_models = 'clad_zrycreep clad_plasticity'
    relative_tolerance = 1e-5
    block = clad
  []
  [clad_zrycreep]
    type = ZryCreepLimbackHoppeUpdate
    block = clad
    temperature = temp
    fast_neutron_flux = fast_neutron_flux
    fast_neutron_fluence = fast_neutron_fluence
    model_irradiation_creep = true
    model_primary_creep = true
    model_thermal_creep = true
    relative_tolerance = 1e-5
    max_inelastic_increment = 1e-4
    zircaloy_material_type = stress_relief_annealed
  []
  [thermal_expansion]
    type = ZryThermalExpansionMATPROEigenstrain
    block = clad
    temperature = temp
    stress_free_temperature = 295.0
    eigenstrain_name = clad_thermal_eigenstrain
  []
  [irradiation_swelling]
    type = ZryIrradiationGrowthEigenstrain
    block = clad
    fast_neutron_fluence = fast_neutron_fluence
    zircaloy_material_type = stress_relief_annealed
    eigenstrain_name = clad_irradiation_strain
  []
  [clad_plasticity]
    type = ZryPlasticityUpdate
    block = clad
    temperature = temp
    fast_neutron_flux = fast_neutron_flux
    fast_neutron_fluence = fast_neutron_fluence
    relative_tolerance = 1e-5
    cold_work_factor = 0.5
    plasticity_model_type = MATPRO
    zircaloy_alloy_type = 4
  []
  [fission_gas_behavior]
    type = U3Si2Sifgrs
    block = pellet_type_1
    temperature = temp
    burnup_function = burnup
    saturation_coverage = 0.5
  []
  [clad_density]
    type = StrainAdjustedDensity
    block = clad
    strain_free_density = 6511.0
  []
  [fuel_density]
    type = StrainAdjustedDensity
    block = pellet_type_1
    strain_free_density = ${initial_fuel_density}
  []
  [ZryOxidation]
    type = ZryOxidation
    boundary = 2
    clad_inner_radius = 4.1783e-3
    clad_outer_radius = 4.7498e-3
    normal_operating_temperature_model = epri_kwu_ce
    temperature = temp
    fast_neutron_flux = fast_neutron_flux
    use_coolant_channel = true
    oxygen_weight_fraction_initial = 0.0012
  []
[]

[Dampers]
  [limitT]
    type = MaxIncrement
    max_increment = 100.0
    variable = temp
  []
  [limitX]
    type = MaxIncrement
    max_increment = 1e-5
    variable = disp_x
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'

  line_search = 'none'

  l_max_its = 100
  l_tol = 8e-3

  nl_max_its = 25
  nl_rel_tol = 1e-5
  nl_abs_tol = 1e-10

  start_time = -200
  n_startup_steps = 1
  end_time = 1e8

  dtmax = 1e6
  dtmin = 1e-3

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 2.0e2
    force_step_every_function_point = true
    timestep_limiting_function = power_history
    max_function_change = 3e20
    optimal_iterations = 10
    iteration_window = 2
    linear_iteration_ratio = 100
    timestep_limiting_postprocessor = material_timestep
  []
  [Quadrature]
    order = FIFTH
    side_order = SEVENTH
  []
[]

[Postprocessors]
  [avg_fuel_surface]
    type = SideAverageValue
    boundary = 10
    variable = temp
  []
  [ave_temp_interior]
    type = SideAverageValue
    boundary = 9
    variable = temp
    execute_on = 'initial linear'
  []
  [clad_inner_vol]
    type = InternalVolume
    boundary = 7
  []
  [pellet_volume]
    type = InternalVolume
    boundary = 8
  []
  [avg_clad_temp]
    type = SideAverageValue
    boundary = 7
    variable = temp
  []
  [fis_gas_produced]
    type = ElementIntegralFisGasGeneratedSifgrs
    block = pellet_type_1
  []
  [fis_gas_released]
    type = ElementIntegralFisGasReleasedSifgrs
    block = pellet_type_1
  []
  [fis_gas_grain]
    type = ElementIntegralFisGasGrainSifgrs
    block = pellet_type_1
  []
  [fis_gas_boundary]
    type = ElementIntegralFisGasBoundarySifgrs
    block = pellet_type_1
  []
  [gas_volume]
    type = InternalVolume
    boundary = 9
    execute_on = 'initial linear'
  []
  [flux_from_clad]
    type = SideDiffusiveFluxIntegral
    variable = temp
    boundary = 5
    diffusivity = thermal_conductivity
  []
  [flux_from_fuel]
    type = SideDiffusiveFluxIntegral
    variable = temp
    boundary = 10
    diffusivity = thermal_conductivity
  []
  [_dt]
    type = TimestepSize
  []
  [num_lin_it]
    type = NumLinearIterations
  []
  [num_nonlin_it]
    type = NumNonlinearIterations
  []
  [tot_lin_it]
    type = CumulativeValuePostprocessor
    postprocessor = num_lin_it
  []
  [tot_nonlin_it]
    type = CumulativeValuePostprocessor
    postprocessor = num_nonlin_it
  []
  [alive_time]
    type = PerfGraphData
    section_name = Root
    data_type = TOTAL
  []
  [rod_total_power]
    type = ElementIntegralPower
    variable = temp
    burnup_function = burnup
    block = pellet_type_1
  []
  [rod_input_power]
    type = FunctionValuePostprocessor
    function = power_history
    scale_factor = 0.1186
  []
  [average_burnup]
    type = ElementAverageValue
    block = pellet_type_1
    variable = burnup
  []
  [oxide_thickness]
    type = ElementExtremeValue
    block = clad
    variable = oxide_thickness
  []
  [fis_gas_percent]
    type = FGRPercent
    fission_gas_released = fis_gas_released
    fission_gas_generated = fis_gas_produced
  []
  [material_timestep]
    type = MaterialTimeStepPostprocessor
    block = clad
  []
[]

[Outputs]
  perf_graph = true
  time_step_interval =  1
  exodus = true
  color = false
  csv = true
  print_linear_residuals = true
  [console]
    type = Console
    max_rows = 25
  []
[]

(test/tests/sifgrs/u3si2/intergranular.i)

# This test verifies the overall U3Si2 fgb model through a single element calculation.
# The outcome has been benchmarked against the result of a stand-alone, independent software.
# In this input file, the coupled fgr and gaseous swelling calculation is performed.

initial_fuel_density = 11590.0

[GlobalParams]
  density = ${initial_fuel_density}
  order = FIRST
  family = LAGRANGE
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 3
    xmin = 0
    xmax = 0.01
    ymin = 0
    ymax = 0.01
    zmin = 0
    zmax = 0.01
    nx = 1
    ny = 1
    nz = 1
  []
[]

[Functions]
  [Temp_func]
    type = ParsedFunction
    expression = '1250'
  []
  [Fiss_func]
    type = ParsedFunction
    expression = '2.e19'
  []
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
  [disp_z]
  []
  [T]
    initial_condition = 1250.0
  []
[]

[AuxVariables]
  [fission_rate]
    order = FIRST
    family = LAGRANGE
  []
  [burnup]
    order = CONSTANT
    family = MONOMIAL
  []
  [rad_bbl_grn]
    order = CONSTANT
    family = MONOMIAL
  []
  [bbl_grn_3]
    order = CONSTANT
    family = MONOMIAL
  []
  [gas_bbl_grn]
    order = CONSTANT
    family = MONOMIAL
  []
  [gas_bdr_3]
    order = CONSTANT
    family = MONOMIAL
  []
  [deltav_v0_intra_total]
    order = CONSTANT
    family = MONOMIAL
  []
  [deltav_v0_bubble_GB]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [fuel]
    strain = FINITE
    eigenstrain_names = 'fuel_thermal_strain fuel_volumetric_strain'
    temperature = T
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable = fission_rate
    value = 1
    fission_rate_function = Fiss_func
    execute_on = 'initial timestep_begin'
  []
  [burnup]
    type = BurnupAux
    variable = burnup
    fission_rate = fission_rate
  []
  [bbl_cnc]
    type = MaterialRealAux
    variable = bbl_grn_3
    property = bubble_concentration_intra
  []
  [rad_bbl]
    type = MaterialRealAux
    variable = rad_bbl_grn
    property = bubble_radius_intra
  []
  [gascnc_bbl]
    type = MaterialRealAux
    variable = gas_bbl_grn
    property = gas_concentration_bubble_intra
  []
  [dvv0gr]
    type = MaterialRealAux
    variable = deltav_v0_intra_total
    property = deltav_v0_intra_total
  []
  [dvv0bd]
    type = MaterialRealAux
    variable = deltav_v0_bubble_GB
    property = deltav_v0_bubble_GB
  []
[]

[BCs]
  [bottom_y]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0.0
  []
  [left_x]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  []
  [back_z]
    type = DirichletBC
    variable = disp_z
    boundary = back
    value = 0.0
  []
  [bottom_T]
    type = FunctionDirichletBC
    variable = T
    function = Temp_func
    boundary = 1
  []
[]

[Materials]
  [U3Si2]
    type = GenericConstantMaterial
    block = 0
    prop_names = "thermal_conductivity specific_heat"
    prop_values = '1 1'
  []
  [fission_gas_behavior]
    type = U3Si2Sifgrs
    block = 0
    temperature = T
    fission_rate = fission_rate
    grain_radius_const = 28.e-06
  []
  [swelling]
    type = U3Si2VolumetricSwellingEigenstrain
    block = 0
    gaseous_swelling_type = U3SI2FG
    temperature = T
    burnup = burnup
    complete_burnup = 10
    eigenstrain_name = fuel_volumetric_strain
  []
  [elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 1.0e11
    poissons_ratio = 0.17
    block = 0
  []
  [elastic_stress]
    type = ComputeFiniteStrainElasticStress
    block = 0
  []
  [thermal_expansion]
    type = ComputeThermalExpansionEigenstrain
    block = 0
    temperature = T
    stress_free_temperature = 1250.0
    thermal_expansion_coeff = 1.5e-5
    eigenstrain_name = fuel_thermal_strain
  []
  [density]
    type = StrainAdjustedDensity
    block = 0
    strain_free_density = ${initial_fuel_density}
  []
[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'

  l_tol = 1e-4
  nl_abs_tol = 1e-7
  nl_rel_tol = 1e-7
  start_time = 0.
  dt = 1e5
  num_steps = 100
[]

[Postprocessors]
  [fis_gas_generated]
    type = ElementIntegralFisGasGeneratedSifgrs
    block = 0
  []
  [fis_gas_released]
    type = ElementIntegralFisGasReleasedSifgrs
    block = 0
  []
  [gbswelling]
    type = ElementAverageValue
    variable = deltav_v0_bubble_GB
    block = 0
  []
  [igswelling]
    type = ElementAverageValue
    variable = deltav_v0_intra_total
    block = 0
  []
[]

[Outputs]
  exodus = true
[]

(test/tests/sifgrs/u3si2/polypole2.i)

# This input is used to test polypole2 with U3Si2Sifgrs

[Mesh]
  [gen]
    type = ExamplePatchMeshGenerator
    dim = 3
  []
[]

[Functions]
  [Temp_func]
    type = ParsedFunction
    expression = '1700'
  []
  [Fiss_func]
    type = ParsedFunction
    expression = '2.5e19'
  []
[]

[Variables]
  [T]
    initial_condition = 1700
  []
[]

[AuxVariables]
  [fission_rate]
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
    diffusion_coefficient = 1
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable = fission_rate
    value = 1
    fission_rate_function = Fiss_func
    execute_on = 'initial timestep_begin'
  []
[]

[BCs]
  [bottom_T]
    type = FunctionDirichletBC
    variable = T
    function = Temp_func
    boundary = 1
  []
[]

[Materials]
  [fission_gas_behavior]
    type = U3Si2Sifgrs
    skip_bdr_model = true
    temperature = T
    fission_rate = fission_rate
    ig_diff_algorithm = POLYPOLE2
    ig_bubble_model = NUCLEATION_RESOLUTION
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  l_tol = 1e-4
  nl_abs_tol = 1e-5
  nl_rel_tol = 1e-5
  start_time = 0.0
  num_steps = 50
  dt = 1e6
[]

[Postprocessors]
  [fis_gas_generated]
    type = ElementIntegralFisGasGeneratedSifgrs
    block = 1
  []
  [fis_gas_released]
    type = ElementIntegralFisGasReleasedSifgrs
    block = 1
  []
  [fgr_percent]
    type = FGRPercent
    fission_gas_released = fis_gas_released
    fission_gas_generated = fis_gas_generated
  []
[]

[Outputs]
  csv = true
  exodus = true
[]

(test/tests/sifgrs/u3si2/option_base.i)

# This base input file is used to test the various input model options

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
  []
[]

[Problem]
  solve = false
[]

[AuxVariables]
  [fission_rate]
    initial_condition = 2.5e19
  []
  [T]
    initial_condition = 1000
  []
[]

[AuxKernels]
  [tempaux]
    type = ParsedAux
    variable = T
    use_xyzt = true
    expression = '1000 + 1000 * t / 50e6'
  []
[]

[Materials]
  [fission_gas_behavior]
    type = U3Si2Sifgrs
    skip_bdr_model = true
    temperature = T
    fission_rate = fission_rate
    ig_diff_algorithm = POLYPOLE2
  []
[]

[Executioner]
  type = Transient
  num_steps = 50
  dt = 1e6
[]

[Postprocessors]
  [fis_gas_generated]
    type = ElementIntegralFisGasGeneratedSifgrs
  []
  [fis_gas_released]
    type = ElementIntegralFisGasReleasedSifgrs
  []
  [fgr_percent]
    type = FGRPercent
    fission_gas_released = fis_gas_released
    fission_gas_generated = fis_gas_generated
  []
[]

[Outputs]
  csv = true
[]

(examples/accident_tolerant_fuel/u3si2_sic/u3si2_outer_monolith_1.5D.i)

# Model is of a 10 pellet fuel rodlet modeled in 1.5D.  The rodlet contains
# U3Si2 fuel and a multilayer silicon carbide cladding (an inner composite
# winding layer) and an outer monolithic layer. The inner composite layer is
# 0.75 mm thick and the outer monolithic layer is 0.25 mm thick. The internal
# layered1D mesh generator can model a clad an arbitrary number of additional blocks.
# Therefore, to create the multilayer SiC clad the composite layer is assigned to the
# clad block and the monolithic_layer is assigned to the monolithic_layer block
# as specified in the additional_block_names parameter in the Mesh block.


initial_fuel_density = 11590.0

[GlobalParams]
  density = ${initial_fuel_density}
  initial_porosity = 0.05
  order = SECOND
  family = LAGRANGE
  energy_per_fission = 3.2e-11 # J/fission
  displacements = disp_x
  temperature = temperature
[]

[Problem]
  type = ReferenceResidualProblem
  reference_vector = 'ref'
  extra_tag_vectors = 'ref'
[]

[Mesh]
  coord_type = RZ
  [layered1D_mesh]
    type = Layered1DMeshGenerator
    slices_per_block = 10
    clad_gap_width = 8.0e-5
    clad_mesh_density = customize
    clad_thickness = 0.00075
    nx_c = 5
    additional_block_names = 'monolithic_layer'
    additional_elements_per_ring = '3'
    additional_ring_thicknesses = '0.00025'
    fuel_height = 0.1186
    plenum_height = 0.027
  []
  patch_update_strategy = auto
  partitioner = centroid
  centroid_partitioner_direction = y
[]

[UserObjects]
  [pin_geometry]
    type = Layered1DFuelPinGeometry
    mesh_generator = layered1D_mesh
  []
[]

[Variables]
  [temperature]
    initial_condition = 580.0 # set initial temperature to coolant inlet
  []
[]

[AuxVariables]
  [disp_y] ## Required for easier visualization in Paraview
  []
  [disp_z] ## Required for easier visualization in Paraview
  []
  [fast_neutron_flux]
    block = 'clad monolithic_layer'
    order = CONSTANT
    family = MONOMIAL
  []
  [fast_neutron_fluence]
    block = 'clad monolithic_layer'
    order = CONSTANT
    family = MONOMIAL
  []
  [grain_radius]
    block = fuel
    initial_condition = 10e-6
  []
  [solid_swell]
    order = CONSTANT
    family = MONOMIAL
    block = fuel
  []
  [gaseous_swelling]
    order = CONSTANT
    family = MONOMIAL
    block = fuel
  []
  [densification]
    order = CONSTANT
    family = MONOMIAL
    block = fuel
  []
  [volumetric_swelling_strain]
    order = CONSTANT
    family = MONOMIAL
    block = fuel
  []
  [relocation]
    order = CONSTANT
    family = MONOMIAL
    block = fuel
  []
[]

[Functions]
  [power_history]
    type = PiecewiseLinear
    x = '0 1e4 1e8'
    y = '0 25000 25000'
    scale_factor = 1
  []
  [axial_peaking_factors]
    type = ParsedFunction
    expression = 1
  []
  [pressure_ramp]
    type = PiecewiseLinear
    x = '-200 0 1e8'
    y = '6.537e-3 1 1'
  []
  [q]
    type = CompositeFunction
    functions = 'power_history axial_peaking_factors'
  []
  [clad_axial_pressure]
    type = CladdingAxialPressureFunction
    plenum_pressure = plenum_pressure
    coolant_pressure = pressure_ramp
    coolant_pressure_scaling_factor = 15.5e6
    fuel_pin_geometry = pin_geometry
  []
  [fuel_axial_pressure]
    type = ParsedFunction
    expression = plenum_pressure
    symbol_names = plenum_pressure
    symbol_values = plenum_pressure
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temperature
    extra_vector_tags = 'ref'
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temperature
    extra_vector_tags = 'ref'
  []
  [heat_source]
    type = NeutronHeatSource
    variable = temperature
    block = fuel
    burnup_function = burnup
    extra_vector_tags = 'ref'
  []
[]

[Physics]
  [SolidMechanics]
    [Layered1D]
      [fuel]
        block = fuel
        add_variables = true
        strain = SMALL
        incremental = true
        add_scalar_variables = true
        out_of_plane_strain_name = strain_yy
        fuel_pin_geometry = pin_geometry
        out_of_plane_pressure_function = fuel_axial_pressure
        eigenstrain_names = 'fuel_thermal_strain fuel_swelling_strain'
        generate_output = 'stress_xx stress_yy stress_zz vonmises_stress strain_xx'
        extra_vector_tags = 'ref'
        group_scalar_vars_in_reference_residual = true
        mesh_generator = layered1D_mesh
      []
      [composite]
        block = clad
        add_variables = true
        strain = SMALL
        incremental = true
        add_scalar_variables = true
        out_of_plane_strain_name = strain_yy
        fuel_pin_geometry = pin_geometry
        out_of_plane_pressure_function = clad_axial_pressure
        eigenstrain_names = 'composite_thermal_strain composite_swelling_strain'
        generate_output = 'stress_xx stress_yy stress_zz vonmises_stress strain_xx'
        extra_vector_tags = 'ref'
        group_scalar_vars_in_reference_residual = true
        mesh_generator = layered1D_mesh
      []
      [monolith]
        block = monolithic_layer
        add_variables = true
        strain = SMALL
        incremental = true
        add_scalar_variables = true
        out_of_plane_strain_name = strain_yy
        fuel_pin_geometry = pin_geometry
        out_of_plane_pressure_function = clad_axial_pressure
        eigenstrain_names = 'monolith_thermal_strain monolith_swelling_strain'
        generate_output = 'stress_xx stress_yy stress_zz vonmises_stress strain_xx'
        extra_vector_tags = 'ref'
        group_scalar_vars_in_reference_residual = true
        mesh_generator = layered1D_mesh
      []
    []
  []
[]

[Burnup]
  [burnup]
    block = fuel
    rod_ave_lin_pow = power_history
    axial_power_profile = axial_peaking_factors
    num_radial = 80
    num_axial = 11
    order = CONSTANT
    family = MONOMIAL
    fuel_pin_geometry = pin_geometry
    fuel_volume_ratio = 1.0
    RPF = RPF
    fuel_type = U3Si2
  []
[]

[AuxKernels]
  [fast_neutron_flux]
    type = MaterialRealAux
    variable = fast_neutron_flux
    property = fast_neutron_flux
    block = 'clad monolithic_layer'
    execute_on = timestep_begin
  []
  [fast_neutron_fluence]
    type = MaterialRealAux
    variable = fast_neutron_fluence
    property = fast_neutron_fluence
    block = 'clad monolithic_layer'
    execute_on = timestep_begin
  []
  [grain_radius]
    type = GrainRadiusAux
    block = fuel
    variable = grain_radius
    temperature = temperature
    execute_on = linear
  []
  [solid_swell]
    type = MaterialRealAux
    variable = solid_swell
    property = solid_swelling
    execute_on = timestep_end
    block = fuel
  []
  [gas_swell]
    type = MaterialRealAux
    variable = gaseous_swelling
    property = gaseous_swelling
    execute_on = timestep_end
    block = fuel
  []
  [densification]
    type = MaterialRealAux
    variable = densification
    property = densification
    execute_on = timestep_end
    block = fuel
  []
  [volumetric_swelling_strain]
    type = MaterialRealAux
    variable = volumetric_swelling_strain
    property = volumetric_swelling_strain
    execute_on = timestep_end
    block = fuel
  []
[]

[Contact]
  [pellet_clad_mechanical]
    primary = 5
    secondary = 10
    formulation = kinematic
    model = frictionless
    penalty = 1e7
  []
[]

[ThermalContact]
  [thermal_contact]
    type = GasGapHeatTransfer
    variable = temperature
    primary = 5
    secondary = 10
    initial_moles = initial_moles # coupling to a postprocessor which supplies the initial plenum/gap gas mass
    gas_released = fis_gas_released # coupling to a postprocessor which supplies the fission gas addition
    contact_pressure = contact_pressure
  []
[]

[BCs]
  [no_x_all] # pin pellets and clad along axis of symmetry (y)
    type = DirichletBC
    variable = disp_x
    boundary = 12
    value = 0.0
  []
  [Pressure] # apply coolant pressure on clad outer walls
    [coolantPressure]
      use_displaced_mesh = false
      boundary = 2
      function = pressure_ramp # use the pressure_ramp function defined above
      factor = 15.5e6
    []
  []
  [PlenumPressure] # apply plenum pressure on clad inner walls and pellet surfaces
    [plenumPressure]
      use_displaced_mesh = false
      boundary = 9
      initial_pressure = 2.0e6
      startup_time = 0
      R = 8.314
      output_initial_moles = initial_moles # coupling to post processor to get initial fill gas mass
      temperature = ave_temp_interior # coupling to post processor to get gas temperature approximation
      volume = gas_volume # coupling to post processor to get gas volume
      material_input = fis_gas_released # coupling to post processor to get fission gas added
      output = plenum_pressure # coupling to post processor to output plenum/gap pressure
    []
  []
[]

[CoolantChannel]
  [convective_clad_surface] # apply convective boundary to clad outer surface
    variable = temperature
    boundary = 2
    inlet_temperature = 580 # K
    inlet_pressure = 15.5e6 # Pa
    inlet_massflux = 3800 # kg/m^2-sec
    rod_diameter = 10.368e-3 # m
    rod_pitch = 1.26e-2 # m
    linear_heat_rate = power_history
    axial_power_profile = axial_peaking_factors
  []
[]

[Materials]
  [flux]
    type = FastNeutronFlux
    calculate_fluence = true
    block = 'clad monolithic_layer'
    rod_ave_lin_pow = power_history
    axial_power_profile = axial_peaking_factors
    factor = 3e13
  []
  ### U3Si2 Fuel
  [fuel_thermal]
    type = SilicideFuelThermal
    block = fuel
    thermal_conductivity_model = WHITE
    temperature = temperature
  []
  [fuel_elasticity_tensor]
    type = U3Si2ElasticityTensor
    block = fuel
  []
  [fuel_stress]
    type = ComputeMultipleInelasticStress
    block = fuel
    tangent_operator = elastic
    inelastic_models = 'fuel_creep'
  []
  [fuel_creep]
    type = U3Si2CreepUpdate
    block = fuel
    temperature = temperature
  []
  [fuel_thermal_expansion]
    type = U3Si2ThermalExpansionEigenstrain
    block = fuel
    temperature = temperature
    stress_free_temperature = 295.0
    eigenstrain_name = fuel_thermal_strain
  []
  [fuel_volumetric_swelling]
    type = U3Si2VolumetricSwellingEigenstrain
    block = fuel
    gaseous_swelling_type = FINLAY
    temperature = temperature
    burnup_function = burnup
    eigenstrain_name = fuel_swelling_strain
  []
  [fission_gas_release]
    type = U3Si2Sifgrs
    block = fuel
    temperature = temperature
    burnup_function = burnup
    grain_radius_const = 2.5e-05
  []

  [fuel_density]
    type = StrainAdjustedDensity
    block = fuel
    strain_free_density = ${initial_fuel_density}
  []

  ### Composite SiC
  [composite_thermal]
    type = CompositeSiCThermal
    thermal_conductivity_model = STONE
    temperature = temperature
    block = clad
  []
  [composite_density]
    type = StrainAdjustedDensity
    block = clad
    strain_free_density = 2700.0
  []
  [composite_elasticity_tensor]
    type = CompositeSiCElasticityTensor
    block = clad
  []
  [composite_stress]
    type = ComputeStrainIncrementBasedStress
    block = clad
  []
  [composite_thermal_expansion]
    type = CompositeSiCThermalExpansionEigenstrain
    block = clad
    stress_free_temperature = 295.0
    temperature = temperature
    eigenstrain_name = composite_thermal_strain
  []
  [composite_irradiation_swelling]
    type = CompositeSiCVolumetricSwellingEigenstrain
    block = clad
    temperature = temperature
    fast_neutron_fluence = fast_neutron_fluence
    swelling_model = KATOH
    number_of_substeps = 1000
    eigenstrain_name = composite_swelling_strain
  []

  ### Monolithic SiC
  [monolith_thermal]
    type = MonolithicSiCThermal
    temperature = temperature
    thermal_conductivity_model = STONE
    block = monolithic_layer
  []
  [monolith_density]
    type = StrainAdjustedDensity
    block = monolithic_layer
    strain_free_density = 3120.0
  []
  [monolith_elasticity_tensor]
    type = MonolithicSiCElasticityTensor
    block = monolithic_layer
  []
  [monolith_stress]
    type = ComputeMultipleInelasticStress
    block = monolithic_layer
    tangent_operator = elastic
    inelastic_models = 'monolith_creep'
  []
  [monolith_creep]
    type = MonolithicSiCCreepUpdate
    block = monolithic_layer
    fast_neutron_flux = fast_neutron_flux
    temperature = temperature
    k_function = 2e-37
  []
  [monolith_thermal_expansion]
    type = MonolithicSiCThermalExpansionEigenstrain
    block = monolithic_layer
    stress_free_temperature = 295.0
    temperature = temperature
    eigenstrain_name = monolith_thermal_strain
  []
  [monolith_irradiation_swelling]
    type = CompositeSiCVolumetricSwellingEigenstrain
    block = monolithic_layer
    temperature = temperature
    fast_neutron_fluence = fast_neutron_fluence
    swelling_model = KATOH
    number_of_substeps = 1000
    eigenstrain_name = monolith_swelling_strain
  []
[]

[Dampers]
  [limitT]
    type = MaxIncrement
    max_increment = 100.0
    variable = temperature
  []
  [limitX]
    type = MaxIncrement
    max_increment = 1e-5
    variable = disp_x
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu superlu_dist'

  line_search = 'none'

  l_max_its = 50
  l_tol = 8e-3
  nl_max_its = 50
  nl_rel_tol = 1e-4
  nl_abs_tol = 1e-7

  start_time = -200
  n_startup_steps = 1
  end_time = 8.0e7
  dtmax = 1e6
  dtmin = 1

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 2e2
    optimal_iterations = 25
    iteration_window = 5
    growth_factor = 2
    cutback_factor = .5
  []
[]

[Postprocessors]
  [ave_temp_interior] # average temperature of the cladding interior and all pellet exteriors
    type = LayeredSideAverageValuePostprocessor
    boundary = 9
    variable = temperature
    execute_on = 'initial linear'
    fuel_pin_geometry = pin_geometry
  []
  [clad_inner_vol] # volume inside of cladding
    type = LayeredInternalVolumePostprocessor
    boundary = 7
    component = 0
    fuel_pin_geometry = pin_geometry
    out_of_plane_strain = strain_yy
    execute_on = 'initial linear'
  []
  [pellet_volume] # fuel pellet total volume
    type = LayeredInternalVolumePostprocessor
    boundary = 8
    component = 0
    fuel_pin_geometry = pin_geometry
    out_of_plane_strain = strain_yy
    execute_on = 'initial linear'
  []
  [avg_clad_temp] # average temperature of cladding interior
    type = LayeredSideAverageValuePostprocessor
    boundary = 7
    variable = temperature
    fuel_pin_geometry = pin_geometry
    execute_on = 'initial linear'
  []
  [fis_gas_produced] # fission gas produced (moles)
    type = LayeredElementIntegralFisGasGeneratedSifgrsPostprocessor
    block = fuel
    fuel_pin_geometry = pin_geometry
  []
  [fis_gas_released] # fission gas released to plenum (moles)
    type = LayeredElementIntegralFisGasReleasedSifgrsPostprocessor
    block = fuel
    fuel_pin_geometry = pin_geometry
  []
  [fis_gas_grain]
    type = LayeredElementIntegralFisGasGrainSifgrsPostprocessor
    block = fuel
    outputs = exodus
    fuel_pin_geometry = pin_geometry
  []
  [fis_gas_boundary]
    type = LayeredElementIntegralFisGasBoundarySifgrsPostprocessor
    block = fuel
    outputs = exodus
    fuel_pin_geometry = pin_geometry
  []
  [fission_gas_release]
    type = FGRPercent
    fission_gas_released = fis_gas_released
    fission_gas_generated = fis_gas_produced
  []

  [gas_volume]
    type = LayeredInternalVolumePostprocessor
    boundary = 9
    execute_on = 'initial linear'
    component = 0
    out_of_plane_strain = strain_yy
    fuel_pin_geometry = pin_geometry
  []
  [flux_from_clad] # area integrated heat flux from the cladding
    type = LayeredSideFluxIntegralPostprocessor
    variable = temperature
    boundary = 5
    diffusivity = thermal_conductivity
    fuel_pin_geometry = pin_geometry
  []
  [flux_from_fuel] # area integrated heat flux from the fuel
    type = LayeredSideFluxIntegralPostprocessor
    variable = temperature
    boundary = 10
    diffusivity = thermal_conductivity
    fuel_pin_geometry = pin_geometry
  []
  [_dt] # time step
    type = TimestepSize
  []
  [num_lin_it]
    type = NumLinearIterations
  []
  [num_nonlin_it]
    type = NumNonlinearIterations
  []
  [tot_lin_it]
    type = CumulativeValuePostprocessor
    postprocessor = num_lin_it
  []
  [tot_nonlin_it]
    type = CumulativeValuePostprocessor
    postprocessor = num_nonlin_it
  []
  [alive_time]
    type = PerfGraphData
    section_name = Root
    data_type = TOTAL
  []

  [rod_total_power]
    type = LayeredElementIntegralPowerPostprocessor
    variable = temperature
    burnup_function = burnup
    block = fuel
    fuel_pin_geometry = pin_geometry
  []
  [rod_input_power]
    type = FunctionValuePostprocessor
    function = power_history
    scale_factor = 0.1186 # rod height
  []

  [solid_swelling]
    type = ElementAverageValue
    variable = solid_swell
    block = fuel
  []
  [gaseous_swelling]
    type = ElementAverageValue
    variable = gaseous_swelling
    block = fuel
  []
  [densification]
    type = ElementAverageValue
    variable = densification
    block = fuel
  []
  [volumetric_swelling]
    type = ElementAverageValue
    variable = volumetric_swelling_strain
    block = fuel
  []
[]

[Outputs]
  perf_graph = true
  exodus = true
  csv = true
  color = false
[]

(test/tests/sifgrs/u3si2/burnup_function.i)

# This input tests the coupling of burnup function action to U3Si2Sifgrs

[Mesh]
  [gen]
    type = ExamplePatchMeshGenerator
    dim = 3
  []
[]

[Functions]
  [Temp_func]
    type = ParsedFunction
    expression = '1700'
  []
  [Fiss_func]
    type = ParsedFunction
    expression = '2.5e19'
  []
[]

[Variables]
  [T]
    initial_condition = 1700
  []
[]

[AuxVariables]
  [fission_rate]
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
    diffusion_coefficient = 1
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable = fission_rate
    value = 1
    fission_rate_function = Fiss_func
    execute_on = 'initial timestep_begin'
  []
[]

[BCs]
  [bottom_T]
    type = FunctionDirichletBC
    variable = T
    function = Temp_func
    boundary = 1
  []
[]

[Burnup]
  [burnup]
    block = 2
    rod_ave_lin_pow = 1
    axial_power_profile = 1
    num_radial = 12
    num_axial = 9
    a_upper = 0.01496
    a_lower = 0.00226
    fuel_inner_radius = 0.
    fuel_outer_radius = 0.005305
    fuel_volume_ratio = 1.
    order = CONSTANT
    family = MONOMIAL
    density = 10000
  []
[]

[Materials]
  [fission_gas_behavior]
    type = U3Si2Sifgrs
    skip_bdr_model = true
    temperature = T
    burnup_function = burnup
    ig_diff_algorithm = POLYPOLE2
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  l_tol = 1e-4
  nl_abs_tol = 1e-5
  nl_rel_tol = 1e-5
  start_time = 0.0
  num_steps = 50
  dt = 1e6
[]

[Postprocessors]
  [fis_gas_generated]
    type = ElementIntegralFisGasGeneratedSifgrs
    block = 1
  []
  [fis_gas_released]
    type = ElementIntegralFisGasReleasedSifgrs
    block = 1
  []
  [fgr_percent]
    type = FGRPercent
    fission_gas_released = fis_gas_released
    fission_gas_generated = fis_gas_generated
  []
[]

[Outputs]
  exodus = true
[]

(test/tests/sifgrs/u3si2/polypole2_ext_fsngas.i)

# This input tests external fission gas coupling to U3Si2Sifgrs with PolyPole-2

[Mesh]
  [gen]
    type = ExamplePatchMeshGenerator
    dim = 3
  []
[]

[Functions]
  [Temp_func]
    type = ParsedFunction
    expression = '1700'
  []
  [Fiss_func]
    type = ParsedFunction
    expression = '2.5e19'
  []
[]

[Variables]
  [T]
    initial_condition = 1700
  []
[]

[AuxVariables]
  [fission_rate]
  []
  [ext_gas]
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
    diffusion_coefficient = 1
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable = fission_rate
    value = 1
    fission_rate_function = Fiss_func
    execute_on = 'initial timestep_begin'
  []
  [ext_gas_aux]
    type = ParsedAux
    variable = ext_gas
    coupled_variables = 'fission_rate'
    use_xyzt = true
    expression = 'fission_rate * 0.3 / 6.02214076e23 * t'
  []
[]

[BCs]
  [bottom_T]
    type = FunctionDirichletBC
    variable = T
    function = Temp_func
    boundary = 1
  []
[]

[Materials]
  [fission_gas_behavior]
    type = U3Si2Sifgrs
    skip_bdr_model = true
    temperature = T
    fission_rate = fission_rate
    fission_gas_conc = ext_gas
    ig_diff_algorithm = POLYPOLE2
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  l_tol = 1e-4
  nl_abs_tol = 1e-5
  nl_rel_tol = 1e-5
  start_time = 0.0
  num_steps = 50
  dt = 1e6
[]

[Postprocessors]
  [fis_gas_generated]
    type = ElementIntegralFisGasGeneratedSifgrs
    block = 1
  []
  [fis_gas_released]
    type = ElementIntegralFisGasReleasedSifgrs
    block = 1
  []
  [fgr_percent]
    type = FGRPercent
    fission_gas_released = fis_gas_released
    fission_gas_generated = fis_gas_generated
  []
[]

[Outputs]
  exodus = true
  hide = 'ext_gas'
[]

(test/tests/solid_mechanics/u3si2_eigenstrains/u3si2_vswelling/swelling_mechanistic.i)

# This test verifies the overall U3Si2 fgb model through a single element calculation.
# The outcome has been benchmarked against the result of a stand-alone, independent software.
# In this input file, the coupled fgr and gaseous swelling calculation is performed.

initial_fuel_density = 11590.0

[GlobalParams]
  density = ${initial_fuel_density}
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 3
    xmin = 0
    xmax = 0.01
    ymin = 0
    ymax = 0.01
    zmin = 0
    zmax = 0.01
    nx = 1
    ny = 1
    nz = 1
  []
[]

[Variables]
  [T]
    order = FIRST
    family = LAGRANGE
    initial_condition = 1250.0
  []
[]

[AuxVariables]
  [fission_rate]
    order = FIRST
    family = LAGRANGE
  []
  [burnup]
    order = CONSTANT
    family = MONOMIAL
  []
  [rad_bbl_grn]
    order = CONSTANT
    family = MONOMIAL
  []
  [bbl_grn_3]
    order = CONSTANT
    family = MONOMIAL
  []
  [gas_bbl_grn]
    order = CONSTANT
    family = MONOMIAL
  []
  [gas_bdr_3]
    order = CONSTANT
    family = MONOMIAL
  []
  [deltav_v0_intra_total]
    order = CONSTANT
    family = MONOMIAL
  []
  [deltav_v0_bubble_GB]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = T
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = FINITE
    add_variables = true
    eigenstrain_names = 'fuelthermal_strain volumetric_swelling'
    temperature = T
  []
[]

[Functions]
  [Temp_func]
    type = ParsedFunction
    expression = '1250'
  []
  [Fiss_func]
    type = ParsedFunction
    expression = '2.e19'
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable = fission_rate
    value = 1
    fission_rate_function = Fiss_func
    execute_on = 'initial  timestep_begin'
  []
  [burnup]
    type = BurnupAux
    variable = burnup
    fission_rate = fission_rate
  []
  [bbl_cnc]
    type = MaterialRealAux
    variable = bbl_grn_3
    property = bubble_concentration_intra
  []
  [rad_bbl]
    type = MaterialRealAux
    variable = rad_bbl_grn
    property = bubble_radius_intra
  []
  [gascnc_bbl]
    type = MaterialRealAux
    variable = gas_bbl_grn
    property = gas_concentration_bubble_intra
  []
  [dvv0gr]
    type = MaterialRealAux
    variable = deltav_v0_intra_total
    property = deltav_v0_intra_total
  []
  [dvv0bd]
    type = MaterialRealAux
    variable = deltav_v0_bubble_GB
    property = deltav_v0_bubble_GB
  []
[]

[BCs]
  [bottom_y]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0.0
  []
  [left_x]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  []
  [back_z]
    type = DirichletBC
    variable = disp_z
    boundary = back
    value = 0.0
  []
  [bottom_T]
    type = FunctionDirichletBC
    variable = T
    function = Temp_func
    boundary = bottom
  []
[]

[Materials]
  [fuel_elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    block = 0
    youngs_modulus = 1e11
    poissons_ratio = 0.17
  []
  [fission_gas_behavior]
    type = U3Si2Sifgrs
    block = 0
    temperature = T
    fission_rate = fission_rate
    grain_radius_const = 28.e-06
  []
  [fuel_swelling]
    type = U3Si2VolumetricSwellingEigenstrain
    temperature = T
    burnup = burnup
    complete_burnup = 10
    eigenstrain_name = volumetric_swelling
    gaseous_swelling_type = U3SI2FG
  []
  [fuel_thermal_strain]
    type = ComputeThermalExpansionEigenstrain
    block = 0
    thermal_expansion_coeff = 1.5e-5
    temperature = T
    stress_free_temperature = 1250.0
    eigenstrain_name = fuelthermal_strain
  []
  [fuel_stress]
    type = ComputeFiniteStrainElasticStress
    block = 0
  []
  [U3Si2_thermal]
    type = HeatConductionMaterial
    block = 0
    thermal_conductivity = 1.0
    specific_heat = 1.0
  []
  [density]
    type = StrainAdjustedDensity
    block = 0
    strain_free_density = ${initial_fuel_density}
  []
[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'

  petsc_options = '-snes_ksp_ew '
  petsc_options_iname = '-ksp_gmres_restart -pc_type -pc_hypre_type'
  petsc_options_value = '70 hypre boomeramg'
  l_max_its = 60
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-15
  l_tol = 1e-5

  start_time = 0.0
  dt = 100.
  num_steps = 10
[]

[Postprocessors]
  [fis_gas_generated]
    type = ElementIntegralFisGasGeneratedSifgrs
    block = 0
  []
  [fis_gas_released]
    type = ElementIntegralFisGasReleasedSifgrs
    block = 0
  []
  [gbswelling]
    type = ElementAverageValue
    variable = deltav_v0_bubble_GB
    block = 0
  []
  [igswelling]
    type = ElementAverageValue
    variable = deltav_v0_intra_total
    block = 0
  []
[]

[Outputs]
  exodus = true
[]

Description
Coupled fission gas release and swelling model
Grain - boundary gas behavior
Example Input Syntax
Input Parameters
Input Files
References