{
 "title": "Heat through a tie: two blocks of conductivities 10 and 40 with non-matching meshes, 20 and 100 at the ends",
 "kind": "theory",
 "source": "Series thermal resistances: steady one-dimensional conduction through two layers in perfect contact, T_i = (k1 T0 + k2 T1)/(k1 + k2) and q = ΔT/(L1/k1 + L2/k2) (F. P. Incropera, D. P. DeWitt, Fundamentals of Heat and Mass Transfer, §3.1, the composite wall). A linear field in each layer is the exact discrete solution, and the dual mortar tie reproduces a linear temperature across non-matching meshes",
 "quantities": [
  {
   "id": "t_interface",
   "quantity": "T at the tied secondary node (0.5, 0.5, 1)",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Heat",
     "at": [
      [
       0.5,
       0.5,
       1.0
      ]
     ]
    },
    "pointer": "/nodes/0/temperature"
   },
   "value": 84.0,
   "tolerance": {
    "relative": 1e-09
   },
   "reason": "exact discrete solution: linear in each block, and the tie holds the interface temperature for non-matching meshes"
  },
  {
   "id": "t_lower",
   "quantity": "T in the lower block at (0, 0, 0.5)",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Heat",
     "at": [
      [
       0.0,
       0.0,
       0.5
      ]
     ]
    },
    "pointer": "/nodes/0/temperature"
   },
   "value": 52.0,
   "tolerance": {
    "relative": 1e-09
   },
   "reason": "exact discrete solution: linear in each block, and the tie holds the interface temperature for non-matching meshes"
  },
  {
   "id": "t_upper",
   "quantity": "T in the upper block at (1, 1, 1.5)",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Heat",
     "at": [
      [
       1.0,
       1.0,
       1.5
      ]
     ]
    },
    "pointer": "/nodes/0/temperature"
   },
   "value": 92.0,
   "tolerance": {
    "relative": 1e-09
   },
   "reason": "exact discrete solution: linear in each block, and the tie holds the interface temperature for non-matching meshes"
  },
  {
   "id": "q_lower",
   "quantity": "heat flux q_z at (0, 0, 0.5)",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Heat",
     "at": [
      [
       0.0,
       0.0,
       0.5
      ]
     ]
    },
    "pointer": "/nodes/0/heat_flux/2"
   },
   "value": -640.0,
   "tolerance": {
    "relative": 1e-09
   },
   "reason": "the heat flow ΔT/(1/k1 + 1/k2) = 640 per unit area, −k ∂T/∂z uniform in each block"
  },
  {
   "id": "q_upper",
   "quantity": "heat flux q_z at (1, 1, 1.5)",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Heat",
     "at": [
      [
       1.0,
       1.0,
       1.5
      ]
     ]
    },
    "pointer": "/nodes/0/heat_flux/2"
   },
   "value": -640.0,
   "tolerance": {
    "relative": 1e-09
   },
   "reason": "the same heat flow crosses the tie: nothing is lost at the interface"
  }
 ]
}