Darcy Flow Analysis and Convection Topology Optimization

Forced Convection for Linear Steady-State Heat Transfer is available via Darcy Flow analysis.

Darcy Flow Analysis is currently only supported for steady-state heat transfer analysis. Forced convection applications include cooling solutions for electric motors, machine tools (casting, forming), heat exchangers, HVAC systems, and cooling for electronic devices including PCBs. Additionally, Topology Optimization is available for steady-state heat transfer with Darcy flow analysis. The topology optimization considers the effect of forced convection for cooling in conjunction with structural steady-state heat transfer analysis. Topology optimization can help optimize cooling channel structures and placement for a wide range of applications.

The flow solution is described by:(1) K p p= f p MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4samaaBa aaleaacaWGWbaabeaakiaahchacqGH9aqpcaWHMbWaaSbaaSqaaiaa dchaaeqaaaaa@3C02@
Where,
K p MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4samaaBa aaleaacaWGWbaabeaaaaa@37E9@
Permeability matrix
p MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiCaaaa@36ED@
Nodal pressure in the structure
f p MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOzamaaBa aaleaacaWGWbaabeaaaaa@3804@
Pressure load at the inlet
The fluid flow analysis is solved using Darcy’s Law, which describes the flow of a fluid through a porous medium:(2) u= κ μ p MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaiabg2 da9iabgkHiTmaalaaabaGaeqOUdSgabaGaeqiVd0gaaiabgEGirlaa dchaaaa@3ED8@
Where,
u MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaaaa@36F2@
Fluid velocity
κ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOUdSgaaa@37A6@
Fluid permeability (this is different from thermal conductivity, represented by k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGRbaaaa@39A7@ )
μ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0gaaa@37AA@
Fluid dynamic viscosity
p MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIeTaam iCaaaa@386F@
Pressure differential
The equation can be rewritten as:(3) u e = κ μ B p e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDamaaCa aaleqabaGaamyzaaaakiabg2da9iabgkHiTmaalaaabaGaeqOUdSga baGaeqiVd0gaaiaahkeacaWHWbWaaWbaaSqabeaacaWGLbaaaaaa@4059@
Where,
u e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDamaaCa aaleqabaGaamyzaaaaaaa@3809@
Element fluid velocity
B MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOqaaaa@36BF@
Differential of the shape function
p e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiCamaaCa aaleqabaGaamyzaaaaaaa@3804@
Nodal pressure in the element (which is sourced from the flow solution)
The thermal steady-state heat transfer solution is represented by:(4) K c + C p T = f MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WHlbWaaSbaaSqaaiaadogaaeqaaOGaey4kaSIaaC4qamaabmaabaGa aCiCaaGaayjkaiaawMcaaaGaay5waiaaw2faaiaahsfacqGH9aqpca WHMbaaaa@40DA@
Where,
K c MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4samaaBe aaleaacaWGJbaabeaaaaa@37DD@
Conductivity Matrix: K c = n = 1 N e Ω e k B T B d Ω MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4samaaBa aaleaacaWGJbaabeaakiabg2da9maaqahabaWaa8qeaeaacaWGRbGa aCOqamaaCaaaleqabaGaamivaaaakiaahkeacaWGKbGaeuyQdCfale aacqqHPoWvdaahaaadbeqaaiaadwgaaaaaleqaniabgUIiYdaaleaa caWGUbGaeyypa0JaaGymaaqaaiaad6eadaWgaaadbaGaamyzaaqaba aaniabggHiLdaaaa@4A93@
C p MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4qamaabm aabaGaaCiCaaGaayjkaiaawMcaaaaa@3942@
Convection Matrix (which includes flow velocity u e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDamaaCa aaleqabaGaamyzaaaaaaa@3809@ from Darcy's Law): C p = n = 1 N e Ω e N ^ T ρ c p u e B d Ω MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4qamaabm aabaGaaCiCaaGaayjkaiaawMcaaiabg2da9maaqahabaWaa8qeaeaa ceWHobGbaKaadaahaaWcbeqaaiaadsfaaaGccqaHbpGCcaWGJbWaaS baaSqaaiaadchaaeqaaOGaaCyDamaaCaaaleqabaGaamyzaaaakiaa hkeacaWGKbGaeuyQdCfaleaacqqHPoWvdaahaaadbeqaaiaadwgaaa aaleqaniabgUIiYdaaleaacaWGUbGaeyypa0JaaGymaaqaaiaad6ea daWgaaadbaGaamyzaaqabaaaniabggHiLdaaaa@510E@
Where,
f MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOzaaaa@36E2@
Thermal load vector
T MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCivaaaa@36D1@
Nodal temperature matrix
N ^ T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOtayaaja WaaWbaaSqabeaacaWGubaaaaaa@37E0@
Enhanced shape function
N e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaWGLbaabeaaaaa@37DD@
Total number of elements
ρ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdihaaa@37B4@
Density
k MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E4@
Thermal conductivity (this is different from fluid permeability, Kappa, represented by κ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOUdSgaaa@37A6@ )
c p MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGWbaabeaaaaa@37FD@
Specific heat capacity
u e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDamaaCa aaleqabaGaamyzaaaaaaa@3809@
Element flow velocity from Darcy’s Law

The thermal steady-state solution incorporates forced convection via the Convection Matrix. A topology design space can be defined for a steady-state heat transfer subcase to run the optimization solution which accounts for the forced convection via Darcy flow.

Input

To turn on forced convection flow analysis, for a Steady-State Heat Transfer Analysis subcase, input definition is required for both the thermal structural and flow analysis.

Boundary Conditions

Boundary conditions are required for both thermal structural and fluid flow analysis. The typical structural thermal boundary conditions are available via the SPC Subcase/Bulk Data. For flow analysis, there are two options to define the boundary conditions:
Nodal Pressure
Flow analysis is solved in the same subcase as thermal analysis. The SPCP Subcase Entry and SPCP Bulk Data are available to define flow pressure boundary conditions. Both inlet and outlet flow pressures can be defined using the SPCP entry.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
SPCP SID G D G D G D
Inlet Velocity
Inlet velocity via the INLTVEL Subcase Entry and INLTVEL Bulk Data are alternately available instead of inlet pressure definition via SPCP entries. The outlet pressure still has to be defined using the SPCP entry.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
INLTVEL SID EID VALUE G1 G2 G3 G4

For Darcy flow analysis, using an SPCP Bulk Data/Subcase Entry pair is mandatory to define the outlet flow pressure. However, for inlet velocity, either the INLTVEL Bulk Data/Subcase pair or the SPCP Bulk/Subcase pair can be used. Therefore, the SPCP Subcase Entry can be considered as an entry which turns on flow analysis for a steady-state heat transfer subcase.

Loads

Loading can be applied to either the solid or fluid domain via the typical heat transfer loads. For instance, the SPC entry can be used to define grid temperature loads, the QBDY1 entry can be used to define heat flux loads, and the QVOL entry can be used to define volumetric heat generation loading.

Material Properties

Material Properties for both structure and fluid for forced convection heat transfer analysis can be defined via the MAT4 Bulk Data entry. The structural material properties are typically defined using the first line which specify the fields, K (structural conductivity), CP (structural specific heat), RHO (structural density), and H (convection heat transfer coefficient). H is only used for free convection to ambient in the presence of CONV Bulk Data. For the fluid heat transfer properties, the DARCY continuation line can be used to define KAPPA (fluid permeability), MU (fluid dynamic viscosity), K (fluid conductivity), CP (fluid specific heat), and RHO (fluid density).
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MAT4 MID K CP RHO H   HGEN    
  DARCY KAPPA MU K CP RHO   INELAHTF  

If certain elements are supposed to only be solid elements, then the DARCY continuation line is not required. For elements which reference materials without a DARCY continuation line, then the lowest KAPPA/MU value from all the fluid MAT4 entries is taken and multiplied with 10-9 and this value is used for solid permeability calculations for Darcy flow analysis.

When only Darcy flow analysis is run, without optimization, then each element can reference either a solid-only MAT4 entry (without DARCY continuation line), or a fluid-only MAT4 entry (with DARCY continuation line, but no structural thermal properties).

However, when topology optimization is run, then for elements in the design space, the referenced MAT4 entry should contain both structural and fluid material properties.

Topology Optimization

During topology optimization, each element in the design space can either be a solid (density=1) or a void (fluid, density=0). Therefore, elements in the design space should reference MAT4 entries with both structural and fluid material properties.

For forced convection topology both DOPTPRM,TOPDISC,YES and minimum member size control (either using DOPTPRM,MINDIM or using MEMBSIZ on the DTPL entry) are turned on automatically. If minimum member size control is not turned on by you, then it is automatically activated with a value based on the average mesh size.

The following responses are available for Flow-based Forced Convection Topology Optimization:
  1. Global Thermal Compliance (RTYPE=TCOMP)(5) T c = 1 2 t T f t = 1 2 t T K t + C p t MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGJbaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikda aaGaaCiDamaaCaaaleqabaGaamivaaaakiaahAgadaWgaaWcbaGaam iDaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaahsha daahaaWcbeqaaiaadsfaaaGcdaWadaqaaiaahUeadaWgaaWcbaGaam iDaaqabaGccqGHRaWkcaWHdbWaaeWaaeaacaWHWbaacaGLOaGaayzk aaaacaGLBbGaayzxaaGaaCiDaaaa@4C5F@
  2. Grid Temperature (RTYPE=TEMP)
  3. Nodal Flow Pressure (RTYPE=FLOWPRES)

An example application is that the nodal flow pressure response can be used to constrain the overall pressure drop across the inlet and the outlet at a given inlet velocity. Thereby, the pressure drop for a fluid pump that is used to pump fluid through the structure is constrained. Another way to inherently define the pressure drop value is also to use SPCP Bulk Data to define both the inlet and outlet pressures.

Supported Input

Darcy flow analysis and Convection Topology Optimization is supported for shell and solid elements. The DTPL Bulk Data Entry can be used to turn on Topology Optimization.

Sample Model Setup

The following sample illustrates an example model setup.
$ *****************************************************************
$ DARCY FLOW ANALYSIS – FORCED CONVECTION STEADY STATE HEAT TRANFER
$ *****************************************************************
SUBCASE 1
   SPC = 4	$ Provides Heat transfer boundary conditions or Temperature loading. 
   SPCP = 6 	$ Activates Darcy Flow analysis, while providing outlet pressure
   INLTVEL = 2  $ This is not mandatory. SPCP can also be used to define inlet pressure
   LOAD = 13 	$ Defines Heat Transfer loading via either QBDY1 or QVOL. 
   PRESSURE = ALL $ Turns on nodal pressure output for Darcy flow.  
   VELOCITY = ALL $ Elemental Velocity is output by default for Darcy flow.
BEGIN BULK
$--1---><--2---><--3---><--4---><--5---><--6---><--7---><--8---><--9---><--10-->
QBDY1         13  10000.   10501
MAT4           1 50.2    5.02E8 7.83E-9 1.2765
MAT4           2 50.2    5.02E8 7.83E-9 1.2765    
+        DARCY    0.1    1000.0 0.598   4.183E+9 1.0E-9
INLTVEL        2    9305  1000.0    8605     425  1549  8611
INLTVEL        2   13305  1000.0   12826    8605  8611  12832
INLTVEL        2   17305  1000.0   17047   12826 12832  17053
INLTVEL        2   21305  1000.0   21268   17047 17053  21274
SPC            4     425            0.0
SPC            4     426            0.0
SPCP           6  122612     0.0
SPCP           6  118391     0.0

Output

Generally, any output from Steady-State Heat Transfer analysis, like Grid Temperatures (THERMAL) and Heat Flux (FLUX) are supported for Darcy flow analysis.

In addition, Nodal Flow Pressure (PRESSURE) and Elemental Velocity (VELOCITY) output are available specifically for Darcy flow analysis.

Nodal pressure is a scalar quantity output that is off by default. PRESSURE I/O Entry can be used to turn on the nodal pressure output.

Elemental velocity is a vector output by default, and can be controlled using the VELOCITY I/O Entry.(6) u = κ μ p = κ μ B p e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaiabg2 da9iabgkHiTmaalaaabaGaeqOUdSgabaGaeqiVd0gaaiabgEGirlaa dchacqGH9aqpcqGHsisldaWcaaqaaiabeQ7aRbqaaiabeY7aTbaaca WHcbGaaCiCamaaCaaaleqabaGaamyzaaaaaaa@471E@
Where,
u MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaaaa@36F2@
Elemental velocity
κ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOUdSgaaa@37A6@
Fluid permeability
μ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0gaaa@37AA@
Fluid dynamic viscosity
p MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIeTaam iCaaaa@386F@
Pressure differential
B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOqaaaa@36BE@
Differential of the elemental shape function
p e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiCamaaCa aaleqabaGaamyzaaaaaaa@3804@
Nodal pressures in element e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGLbaaaa@39A1@
The following table illustrates the input and output entries required to run the solution.
Table 1. Darcy Flow Analysis and Convection Topology Optimization Overview
Darcy Flow Analysis and Convection Topology Bulk Data Case Control
Fixed Temperature Boundary Conditions SPC SPC
Fluid Boundary Conditions SPCP, INLTVEL SPCP, INLTVEL
Structural Thermal Material    
Fluid Thermal Material MAT4 (DARCY continuation line)  
Loads SPC (temperature), QBDY1 (flux), QVOL (heat generation) SPC, LOAD
Flow and Structural Heat Transfer Analysis Output   THERMAL, FLUX, PRESSURE, VELOCITY
Optimization DTPL, DRESP1 responses (TCOMP, TEMP, FLOWPRES), DRESP2, DRESP3 DESSUB, DESOBJ, DESGLB