Electrostatic Contact Verification (Two Block Test)

This document describes the two block 1-D verification test for the ElectrostaticContactCondition object. Below is a summary of the test, along with a derivation of the analytic solution used for comparison and the relevant test input file for review.

Summary

A visual summary of the two block verification test domain, as well as relevant boundary and interface conditions is shown below (click to zoom):

Figure 1: Visual summary of the two block verification test with boundary and interface conditions.

It is important to note that in Figure 1:

$var element = document.getElementById("moose-equation-1fa1b2cc-92d5-4571-865c-db944f71e2f1");katex.render("\\sigma_{i}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the electrical conductivity of material $var element = document.getElementById("moose-equation-3d42d94e-de29-4be7-8ce2-2f1992a9c9d5");katex.render("i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ,
$var element = document.getElementById("moose-equation-1734d7bd-fa15-40b7-9291-70d46dfac681");katex.render("C_E", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the electrical contact conductance at the interface, and
$var element = document.getElementById("moose-equation-747b83bc-3cf2-405f-a474-00946613e972");katex.render("\\phi_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the electrostatic potential of material $var element = document.getElementById("moose-equation-e8734a59-9ec4-41dd-8cd8-c968cf938899");katex.render("i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ .

See the ElectrostaticContactCondition documentation for more information about the particular definition of $var element = document.getElementById("moose-equation-24b3c012-b4af-4941-afb2-5cc9483576da");katex.render("C_E", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ . As the ElectrostaticContactCondition object is intended for electrostatic field solves, the PDE being solved within each domain is Poisson's Equation for electrostatic potential. In this case, we are assuming a zero total charge density, which leads to

(1)var element = document.getElementById("moose-equation-f8d422c5-6ed9-4d94-a31e-9845e23b04d3");katex.render("\\nabla \\cdot (\\sigma_i \\nabla \\phi_i) = 0", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Material properties being used in this case are constants in each block, and they are summarized below in Table 1. All material properties were evaluated at a temperature of ~300 K.

Table 1: Material properties for the two block electrostatic contact verification case.

Property (unit)	Value	Source
Stainless Steel (304) Electrical Conductivity (S / m)	$var element = document.getElementById("moose-equation-0c982e28-09a3-44a2-b684-e3c26abcce08");katex.render("1.41867 \\times 10^6", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$	(Cincotti et al., 2007)
Stainless Steel (304) Hardness (Pa)	$var element = document.getElementById("moose-equation-0242ca79-0e16-4d3b-8446-c797b84d8e5a");katex.render("1.92 \\times 10^9", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$	(Cincotti et al., 2007)
Graphite (AT 101) Electrical Conductivity (S / m)	$var element = document.getElementById("moose-equation-22296f50-48cb-4da7-98db-e4b1e799b72d");katex.render("73069.2", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$	(Cincotti et al., 2007)
Graphite (AT 101) Hardness (Pa)	$var element = document.getElementById("moose-equation-a862b3b4-7871-4047-9dc5-8d593e0440a8");katex.render("3.5 \\times 10^9", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$	(Cincotti et al., 2007)

The hardness values shown in Table 1 are used in the ElectrostaticContactCondition object as a harmonic mean of the two values. For reference, the harmonic mean calculation for two values, $var element = document.getElementById("moose-equation-0b36b7d2-84fa-499b-b469-0f4f0a4411c1");katex.render("V_a", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and $var element = document.getElementById("moose-equation-0c8b4557-eb7b-4c96-a7fb-e6260af64a9f");katex.render("V_b", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , is given by

var element = document.getElementById("moose-equation-0142df7b-ea97-46be-9c71-bcdb38df88df");katex.render("V_{Harm} = \\frac{2 V_a V_b}{V_a + V_b}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

In the input file, the harmonic mean of hardness for stainless steel and graphite was calculated and set to be $var element = document.getElementById("moose-equation-85ce82b8-0a81-480e-a3b5-e20454e3f0f6");katex.render("2.4797 \\times 10^9", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ Pa.

Analytic Solution Derivation

In 1-D, Eq. (1) becomes

var element = document.getElementById("moose-equation-c43509d5-900b-49d4-b971-67c3d3c9a795");katex.render("\\sigma_ i \\frac{\\text{d}^2 \\phi_i}{\\text{d} x^2} = 0", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

since $var element = document.getElementById("moose-equation-4528e6cb-4015-4453-bb56-b22107464491");katex.render("\\sigma_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is constant in both domains. This equation means a reasonable guess for a generic solution function for $var element = document.getElementById("moose-equation-8799a420-8ccd-4a49-a547-adf286bf9ad8");katex.render("\\phi_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ would be

var element = document.getElementById("moose-equation-c5786d87-9242-43a4-aacd-2789472e338c");katex.render("\\phi_i(x) = A_i x + C_i", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where $var element = document.getElementById("moose-equation-cbb2c8c4-a12a-4e76-a976-d3163ba2e9c6");katex.render("A_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and $var element = document.getElementById("moose-equation-22d35f57-a310-4cbf-a59d-44bcfbe748ff");katex.render("C_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ are to-be-determined constant coefficients.

Apply boundary conditions

Using the boundary conditions in Figure 1, we can determine $var element = document.getElementById("moose-equation-0170be3f-eea5-437a-b1bc-d81f4ae6d59d");katex.render("C", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ for both the stainless steel and graphite regions:

Stainless Steel

$var element = document.getElementById("moose-equation-9f185dc5-5844-4d9d-8c29-9dc057fa801b");katex.render("\\begin{aligned} \\phi_S (0) &= A_S(0) + C_S \\\\ A_S(0) + C_S &= 1 \\\\ C_S &= 1 \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

Graphite

$var element = document.getElementById("moose-equation-e386fd15-e654-4a0c-8e9f-229a0018a66b");katex.render("\\begin{aligned} \\phi_G (2) &= A_G(2) + C_G \\\\ A_G(2) + C_G &= 0 \\\\ C_G &= -2A_G \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

Apply interface conditions

Now, the interface conditions can be applied from Figure 1. To begin, let's focus on the current density ( $var element = document.getElementById("moose-equation-d119068c-0bcc-4cb8-97e8-1d552fb8d1f9");katex.render("-\\sigma_i \\nabla \\phi_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ) equivalence condition

var element = document.getElementById("moose-equation-3ad484a1-5adb-4497-8938-7681170fb3e2");katex.render("\\sigma_S \\left. \\frac{\\text{d} \\phi_S}{\\text{d} x} \\right|_{x = 1} = \\sigma_G \\left. \\frac{\\text{d} \\phi_G}{\\text{d} x} \\right|_{x = 1}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Taking into account our initial guess for the solution function, this becomes

(2)var element = document.getElementById("moose-equation-42e0d5b4-2f29-4a31-aaed-3747553b6a25");katex.render("\\sigma_S A_S = \\sigma_G A_G", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Next, we can apply the conductance condition from Figure 1, which is

var element = document.getElementById("moose-equation-8a56b403-52eb-4b28-882f-9eecdfe4e9c3");katex.render("\\sigma_S \\left. \\frac{\\text{d} \\phi_S}{\\text{d} x} \\right|_{x = 1} = -C_E (\\phi_S(1) - \\phi_G(1))", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Taking into account our initial guess for the solution function and the constant coefficients solved for above, this becomes

$var element = document.getElementById("moose-equation-90b5bf60-e22f-4bfb-a6c7-a6051ad520fd");katex.render("\\begin{aligned} \\sigma_S A_S &= -C_E (A_S(1) + 1 - A_G (1 - 2)) \\\\ \\sigma_S A_S &= -C_E (A_S + 1 + A_G) \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

(3)var element = document.getElementById("moose-equation-864e0a17-5a72-4b87-8617-535bbbb947e9");katex.render("\\sigma_S A_S = -C_E A_S - C_E - C_E A_G", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Find the remaining coefficients

Using the Elimination Method, we can combine Eq. (2) and Eq. (3) in order to solve for each remaining unknown coefficient. If Eq. (2) is multiplied through on both sides by $var element = document.getElementById("moose-equation-88b59fba-b563-4aa5-b3b0-58c6fcc30a5c");katex.render("C_E", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and Eq. (3) is multiplied through on both sides by $var element = document.getElementById("moose-equation-537d02b9-f629-4d47-be26-8698a884baba");katex.render("\\sigma_G", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , we have

(4)var element = document.getElementById("moose-equation-bdd33486-3c80-41bf-999a-8ee8c1f0a061");katex.render("C_E \\sigma_S A_S - C_E \\sigma_G A_G = 0", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

and

var element = document.getElementById("moose-equation-8ee15a63-af70-41ea-a798-b758485e2c08");katex.render("\\sigma_G \\sigma_S A_S = -C_E \\sigma_G A_S - C_E \\sigma_G - C_E \\sigma_G A_G", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Grouping terms, we then have

(5)var element = document.getElementById("moose-equation-0d8058cd-b161-4e19-9894-8f4e91e802d7");katex.render("(\\sigma_G \\sigma_S + C_E \\sigma_G) A_S + C_E \\sigma_G A_G = -C_E \\sigma_G", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Combining Eq. (4) and Eq. (5) together via addition yields

var element = document.getElementById("moose-equation-0c26783c-b5c2-49da-88da-b93ddbc4709f");katex.render("(\\sigma_G \\sigma_S + C_E \\sigma_G) A_S + C_E \\sigma_S A_S = -C_E \\sigma_G", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

which can then be solved for $var element = document.getElementById("moose-equation-19702585-abc2-4291-8c4c-772e375cd2b2");katex.render("A_S", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

var element = document.getElementById("moose-equation-d1f57153-bc1b-459b-8bbb-ca5cd04a8d27");katex.render("A_S = \\frac{-C_E \\sigma_G}{\\sigma_G \\sigma_S + C_E \\sigma_G + C_E \\sigma_S}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Using Eq. (2), $var element = document.getElementById("moose-equation-64413c50-7333-4892-ad03-e21e81a09c4f");katex.render("A_G", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ can now be solved for

$var element = document.getElementById("moose-equation-887a06ee-fc3b-46b4-8a72-7d777a48329b");katex.render("\\begin{aligned} \\sigma_G A_G &= \\sigma_S \\left[ \\frac{-C_E \\sigma_G}{\\sigma_G \\sigma_S + C_E \\sigma_G + C_E \\sigma_S} \\right] \\\\ A_G &= \\frac{-C_E \\sigma_S}{\\sigma_G \\sigma_S + C_E \\sigma_G + C_E \\sigma_S} \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

Summarize

Now our determined coefficients can be combined to form the complete solutions for both stainless steel and graphite. To summarize, the derived analytical solutions for each domain given the boundary and interface conditions described in Figure 1 is

$var element = document.getElementById("moose-equation-c61b14e8-db43-4a7e-8486-93e811ff6e01");katex.render("\\begin{aligned} \\phi_S (x) &= \\frac{-C_E \\sigma_G}{\\sigma_G \\sigma_S + C_E \\sigma_G + C_E \\sigma_S} x + 1 \\\\ \\phi_G (x) &= \\frac{-C_E \\sigma_S}{\\sigma_G \\sigma_S + C_E \\sigma_G + C_E \\sigma_S} (x - 2) \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

This is implemented in source code as ElectricalContactTestFunc.C and is located within the test source code directory located at modules/electromagnetics/test/src.

Input File

# Regression test for ElectrostaticContactCondition with analytic solution with
# two blocks
#
# dim = 1D
# X = [0,2]
# Interface at X = 1
#
#   stainless_steel        graphite
# +------------------+------------------+
#
# Left BC: Potential = 1
# Right BC: Potential = 0
# Center Interface: ElectrostaticContactCondition
#

[Mesh<<<{"href": "../../../syntax/Mesh/index.html"}>>>]
  [line]
    type = GeneratedMeshGenerator<<<{"description": "Create a line, square, or cube mesh with uniformly spaced or biased elements.", "href": "../../../source/meshgenerators/GeneratedMeshGenerator.html"}>>>
    dim<<<{"description": "The dimension of the mesh to be generated"}>>> = 1
    nx<<<{"description": "Number of elements in the X direction"}>>> = 4
    xmax<<<{"description": "Upper X Coordinate of the generated mesh"}>>> = 2
  []
  [break]
    type = SubdomainBoundingBoxGenerator<<<{"description": "Changes the subdomain ID of elements either (XOR) inside or outside the specified box to the specified ID.", "href": "../../../source/meshgenerators/SubdomainBoundingBoxGenerator.html"}>>>
    input<<<{"description": "The mesh we want to modify"}>>> = line
    block_id<<<{"description": "Subdomain id to set for inside/outside the bounding box"}>>> = 1
    block_name<<<{"description": "Subdomain name to set for inside/outside the bounding box (optional)"}>>> = 'graphite'
    bottom_left<<<{"description": "The bottom left point (in x,y,z with spaces in-between)."}>>> = '1 0 0'
    top_right<<<{"description": "The bottom left point (in x,y,z with spaces in-between)."}>>> = '2 0 0'
  []
  [block_rename]
    type = RenameBlockGenerator<<<{"description": "Changes the block IDs and/or block names for a given set of blocks defined by either block ID or block name. The changes are independent of ordering. The merging of blocks is supported.", "href": "../../../source/meshgenerators/RenameBlockGenerator.html"}>>>
    input<<<{"description": "The mesh we want to modify"}>>> = break
    old_block<<<{"description": "Elements with these block ID(s)/name(s) will be given the new block information specified in 'new_block'"}>>> = 0
    new_block<<<{"description": "The new block ID(s)/name(s) to be given by the elements defined in 'old_block'."}>>> = 'stainless_steel'
  []
  [interface]
    type = SideSetsBetweenSubdomainsGenerator<<<{"description": "MeshGenerator that creates a sideset composed of the nodes located between two or more subdomains.", "href": "../../../source/meshgenerators/SideSetsBetweenSubdomainsGenerator.html"}>>>
    input<<<{"description": "The mesh we want to modify"}>>> = block_rename
    primary_block<<<{"description": "The primary set of blocks for which to draw a sideset between"}>>> = 'stainless_steel'
    paired_block<<<{"description": "The paired set of blocks for which to draw a sideset between"}>>> = 'graphite'
    new_boundary<<<{"description": "The list of boundary names to create on the supplied subdomain"}>>> = 'ssg_interface'
  []
[]

[Variables<<<{"href": "../../../syntax/Variables/index.html"}>>>]
  [potential_graphite]
    block = graphite
  []
  [potential_stainless_steel]
    block = stainless_steel
  []
[]

[AuxVariables<<<{"href": "../../../syntax/AuxVariables/index.html"}>>>]
  [analytic_potential_stainless_steel]
    block = stainless_steel
  []
  [analytic_potential_graphite]
    block = graphite
  []
[]

[Kernels<<<{"href": "../../../syntax/Kernels/index.html"}>>>]
  [electric_graphite]
    type = ADMatDiffusion<<<{"description": "Diffusion equation kernel that takes an isotropic diffusivity from a material property", "href": "../../../source/kernels/ADMatDiffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = potential_graphite
    diffusivity<<<{"description": "The diffusivity value or material property"}>>> = electrical_conductivity
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = graphite
  []
  [electric_stainless_steel]
    type = ADMatDiffusion<<<{"description": "Diffusion equation kernel that takes an isotropic diffusivity from a material property", "href": "../../../source/kernels/ADMatDiffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = potential_stainless_steel
    diffusivity<<<{"description": "The diffusivity value or material property"}>>> = electrical_conductivity
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = stainless_steel
  []
[]

[AuxKernels<<<{"href": "../../../syntax/AuxKernels/index.html"}>>>]
  [analytic_function_aux_stainless_steel]
    type = FunctionAux<<<{"description": "Auxiliary Kernel that creates and updates a field variable by sampling a function through space and time.", "href": "../../../source/auxkernels/FunctionAux.html"}>>>
    function<<<{"description": "The function to use as the value"}>>> = potential_fxn_stainless_steel
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = analytic_potential_stainless_steel
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = stainless_steel
  []
  [analytic_function_aux_graphite]
    type = FunctionAux<<<{"description": "Auxiliary Kernel that creates and updates a field variable by sampling a function through space and time.", "href": "../../../source/auxkernels/FunctionAux.html"}>>>
    function<<<{"description": "The function to use as the value"}>>> = potential_fxn_graphite
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = analytic_potential_graphite
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = graphite
  []
[]

[BCs<<<{"href": "../../../syntax/BCs/index.html"}>>>]
  [elec_left]
    type = ADDirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../source/bcs/ADDirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = potential_stainless_steel
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = left
    value<<<{"description": "Value of the BC"}>>> = 1
  []
  [elec_right]
    type = ADDirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../source/bcs/ADDirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = potential_graphite
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = right
    value<<<{"description": "Value of the BC"}>>> = 0
  []
[]

[InterfaceKernels<<<{"href": "../../../syntax/InterfaceKernels/index.html"}>>>]
  [electric_contact_conductance_ssg]
    type = ElectrostaticContactCondition<<<{"description": "Interface condition that describes the current continuity and contact conductance across a boundary formed between two dissimilar materials (resulting in a potential discontinuity). Conductivity on each side of the boundary is defined via the material properties system.", "href": "../../../source/interfacekernels/ElectrostaticContactCondition.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = potential_stainless_steel
    neighbor_var<<<{"description": "The variable on the other side of the interface."}>>> = potential_graphite
    boundary<<<{"description": "The list of boundaries (ids or names) from the mesh where this object applies"}>>> = ssg_interface
    mean_hardness<<<{"description": "Geometric mean of the hardness of each contacting material."}>>> = mean_hardness
    mechanical_pressure<<<{"description": "Mechanical pressure uniformly applied at the contact surface area (Pressure = Force / Surface Area)."}>>> = 3000
  []
[]

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  #graphite (at 300 K)
  [sigma_graphite]
    type = ADGenericConstantMaterial<<<{"description": "Declares material properties based on names and values prescribed by input parameters.", "href": "../../../source/materials/GenericConstantMaterial.html"}>>>
    prop_names<<<{"description": "The names of the properties this material will have"}>>> = electrical_conductivity
    prop_values<<<{"description": "The values associated with the named properties"}>>> = 73069.2
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = graphite
  []

  #stainless_steel (at 300 K)
  [sigma_stainless_steel]
    type = ADGenericConstantMaterial<<<{"description": "Declares material properties based on names and values prescribed by input parameters.", "href": "../../../source/materials/GenericConstantMaterial.html"}>>>
    prop_names<<<{"description": "The names of the properties this material will have"}>>> = electrical_conductivity
    prop_values<<<{"description": "The values associated with the named properties"}>>> = 1.41867e6
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = stainless_steel
  []

  # harmonic mean of graphite and stainless steel hardness
  [mean_hardness]
    type = ADGenericConstantMaterial<<<{"description": "Declares material properties based on names and values prescribed by input parameters.", "href": "../../../source/materials/GenericConstantMaterial.html"}>>>
    prop_names<<<{"description": "The names of the properties this material will have"}>>> = mean_hardness
    prop_values<<<{"description": "The values associated with the named properties"}>>> = 2.4797e9
  []
[]

[Functions<<<{"href": "../../../syntax/Functions/index.html"}>>>]
  [potential_fxn_stainless_steel]
    type = ElectricalContactTestFunc
    domain = stainless_steel
  []
  [potential_fxn_graphite]
    type = ElectricalContactTestFunc
    domain = graphite
  []
[]

[Postprocessors<<<{"href": "../../../syntax/Postprocessors/index.html"}>>>]
  [error_stainless_steel]
    type = ElementL2Error<<<{"description": "Computes L2 error between a field variable and an analytical function", "href": "../../../source/postprocessors/ElementL2Error.html"}>>>
    variable<<<{"description": "The name of the variable that this object operates on"}>>> = potential_stainless_steel
    function<<<{"description": "The analytic solution to compare against"}>>> = potential_fxn_stainless_steel
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = stainless_steel
  []
  [error_graphite]
    type = ElementL2Error<<<{"description": "Computes L2 error between a field variable and an analytical function", "href": "../../../source/postprocessors/ElementL2Error.html"}>>>
    variable<<<{"description": "The name of the variable that this object operates on"}>>> = potential_graphite
    function<<<{"description": "The analytic solution to compare against"}>>> = potential_fxn_graphite
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = graphite
  []
[]

[Preconditioning<<<{"href": "../../../syntax/Preconditioning/index.html"}>>>]
  [SMP]
    type = SMP<<<{"description": "Single matrix preconditioner (SMP) builds a preconditioner using user defined off-diagonal parts of the Jacobian.", "href": "../../../source/preconditioners/SingleMatrixPreconditioner.html"}>>>
    full<<<{"description": "Set to true if you want the full set of couplings between variables simply for convenience so you don't have to set every off_diag_row and off_diag_column combination."}>>> = true
  []
[]

[Executioner<<<{"href": "../../../syntax/Executioner/index.html"}>>>]
  type = Steady
  solve_type = NEWTON
  automatic_scaling = true
[]

[Outputs<<<{"href": "../../../syntax/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
  perf_graph<<<{"description": "Enable printing of the performance graph to the screen (Console)"}>>> = true
[]

Results

Results from the input file shown above (with Mesh/line/nx=30 and Outputs/exodus=true) compared to the analytic function are shown below in Figure 2. Note that the number of points shown in the plot has been down-sampled compared to the solved number of elements for readability.

Figure 2: Results of electrostatic contact two block validation case.

References

A. Cincotti, A. M. Locci, R. Orrù, and G. Cao. Modeling of SPS apparatus: temperature, current and strain distribution with no powders. AIChE Journal, 53(3):703–719, 2007. doi:10.1002/aic.11102.

@article{cincotti2007sps,
    author = "Cincotti, A. and Locci, A. M. and Orrù, R. and Cao, G.",
    title = "Modeling of {SPS} apparatus: Temperature, current and strain distribution with no powders",
    journal = "AIChE Journal",
    volume = "53",
    number = "3",
    pages = "703-719",
    doi = "10.1002/aic.11102",
    year = "2007"
}

(modules/electromagnetics/test/tests/interfacekernels/electrostatic_contact/analytic_solution_test_two_block.i)

# Regression test for ElectrostaticContactCondition with analytic solution with
# two blocks
#
# dim = 1D
# X = [0,2]
# Interface at X = 1
#
#   stainless_steel        graphite
# +------------------+------------------+
#
# Left BC: Potential = 1
# Right BC: Potential = 0
# Center Interface: ElectrostaticContactCondition
#

[Mesh]
  [line]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 4
    xmax = 2
  []
  [break]
    type = SubdomainBoundingBoxGenerator
    input = line
    block_id = 1
    block_name = 'graphite'
    bottom_left = '1 0 0'
    top_right = '2 0 0'
  []
  [block_rename]
    type = RenameBlockGenerator
    input = break
    old_block = 0
    new_block = 'stainless_steel'
  []
  [interface]
    type = SideSetsBetweenSubdomainsGenerator
    input = block_rename
    primary_block = 'stainless_steel'
    paired_block = 'graphite'
    new_boundary = 'ssg_interface'
  []
[]

[Variables]
  [potential_graphite]
    block = graphite
  []
  [potential_stainless_steel]
    block = stainless_steel
  []
[]

[AuxVariables]
  [analytic_potential_stainless_steel]
    block = stainless_steel
  []
  [analytic_potential_graphite]
    block = graphite
  []
[]

[Kernels]
  [electric_graphite]
    type = ADMatDiffusion
    variable = potential_graphite
    diffusivity = electrical_conductivity
    block = graphite
  []
  [electric_stainless_steel]
    type = ADMatDiffusion
    variable = potential_stainless_steel
    diffusivity = electrical_conductivity
    block = stainless_steel
  []
[]

[AuxKernels]
  [analytic_function_aux_stainless_steel]
    type = FunctionAux
    function = potential_fxn_stainless_steel
    variable = analytic_potential_stainless_steel
    block = stainless_steel
  []
  [analytic_function_aux_graphite]
    type = FunctionAux
    function = potential_fxn_graphite
    variable = analytic_potential_graphite
    block = graphite
  []
[]

[BCs]
  [elec_left]
    type = ADDirichletBC
    variable = potential_stainless_steel
    boundary = left
    value = 1
  []
  [elec_right]
    type = ADDirichletBC
    variable = potential_graphite
    boundary = right
    value = 0
  []
[]

[InterfaceKernels]
  [electric_contact_conductance_ssg]
    type = ElectrostaticContactCondition
    variable = potential_stainless_steel
    neighbor_var = potential_graphite
    boundary = ssg_interface
    mean_hardness = mean_hardness
    mechanical_pressure = 3000
  []
[]

[Materials]
  #graphite (at 300 K)
  [sigma_graphite]
    type = ADGenericConstantMaterial
    prop_names = electrical_conductivity
    prop_values = 73069.2
    block = graphite
  []

  #stainless_steel (at 300 K)
  [sigma_stainless_steel]
    type = ADGenericConstantMaterial
    prop_names = electrical_conductivity
    prop_values = 1.41867e6
    block = stainless_steel
  []

  # harmonic mean of graphite and stainless steel hardness
  [mean_hardness]
    type = ADGenericConstantMaterial
    prop_names = mean_hardness
    prop_values = 2.4797e9
  []
[]

[Functions]
  [potential_fxn_stainless_steel]
    type = ElectricalContactTestFunc
    domain = stainless_steel
  []
  [potential_fxn_graphite]
    type = ElectricalContactTestFunc
    domain = graphite
  []
[]

[Postprocessors]
  [error_stainless_steel]
    type = ElementL2Error
    variable = potential_stainless_steel
    function = potential_fxn_stainless_steel
    block = stainless_steel
  []
  [error_graphite]
    type = ElementL2Error
    variable = potential_graphite
    function = potential_fxn_graphite
    block = graphite
  []
[]


[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  automatic_scaling = true
[]

[Outputs]
  csv = true
  perf_graph = true
[]

Summary
Analytic Solution Derivation
Input File
Results
References