10.1. Coupled Effects

The following topics concerning coupled effects are available:

 

10.1.1. Elements

The following elements have coupled-field capability:

Table 10.1: Elements Used for Coupled Effects

SOLID5 3D Coupled-Field Solid (Derivation of Electromagnetic Matrices, Coupled Effects, SOLID5 - 3D Coupled-Field Solid)
PLANE13 2D Coupled-Field Solid (Derivation of Electromagnetic Matrices, Coupled Effects, SOLID5 - 3D Coupled-Field Solid)
FLUID29 2D Acoustic Fluid (Derivation of Acoustic Matrices, FLUID29 - 2D Acoustic Fluid)
FLUID30 3D 8-Node Acoustic Fluid (Derivation of Acoustic Matrices, FLUID30 - 3D Acoustic Fluid)
LINK68 Coupled Thermal-Electric Line (LINK68 - Coupled Thermal-Electric Line)
SOLID98 Tetrahedral Coupled-Field Solid (Derivation of Electromagnetic Matrices, Coupled Effects, SOLID98 - Tetrahedral Coupled-Field Solid)
FLUID116 Coupled Thermal-Fluid Pipe (FLUID116 - Coupled Thermal-Fluid Pipe)
CIRCU124 Electric Circuit Element (CIRCU124 - Electric Circuit)
TRANS126 Electromechanical Transducer (Capacitance Computation, Review of Coupled Electromechanical Methods, TRANS126 - Electromechanical Transducer)
SHELL157 Coupled Thermal-Electric Shell (SHELL157 - Thermal-Electric Shell)
FLUID220 3D 20-Node Acoustic Fluid
FLUID221 3D 10-Node Acoustic Fluid
PLANE222 2D 4-Node Coupled-Field Solid (PLANE222 - 2D 4-Node Coupled-Field Solid)
PLANE223 2D 8-Node Coupled-Field Solid (PLANE223 - 2D 8-Node Coupled-Field Solid)
SOLID225 3D 8-Node Coupled-Field Solid (SOLID225 - 3D 8-Node Coupled-Field Solid)
SOLID226 3D 20-Node Coupled-Field Solid (SOLID226 - 3D 20-Node Coupled-Field Solid)
SOLID227 3D 10-Node Coupled-Field Solid (SOLID227 - 3D 10-Node Coupled-Field Solid)
LINK2283D Coupled-Field Link (LINK228 - 3D Coupled-Field Link)
PLANE233 2D 8-Node Electromagnetic Solid (Derivation of Electromagnetic Matrices, Electromagnetic Field Evaluations, PLANE233 - 2D 8-Node Electromagnetic Solid)
SOLID236 3D 20-Node Electromagnetic Solid (Derivation of Electromagnetic Matrices, Electromagnetic Field Evaluations, SOLID236 - 3D 20-Node Electromagnetic Solid)
SOLID237 3D 10-Node Electromagnetic Solid (Derivation of Electromagnetic Matrices, Electromagnetic Field Evaluations, SOLID237 - 3D 10-Node Electromagnetic Solid)

There are certain advantages and disadvantages inherent with coupled-field formulations:

 

10.1.1.1. Advantages

  1. Allows for solutions to problems otherwise not possible with usual finite elements.

  2. Simplifies modeling of coupled-field problems by permitting one element type to be used in a single analysis pass.

 

10.1.1.2. Disadvantages

  1. Increases problem size (unless a segregated solver is used).

  2. Inefficient matrix reformulation (if a section of a matrix associated with one phenomena is reformed, the entire matrix will be reformed).

  3. Larger storage requirements.

 

10.1.2. Coupling Methods

There are basically two methods of coupling distinguished by the finite element formulation techniques used to develop the matrix equations. These are illustrated here with two types of degrees of freedom ({X1}, {X2}):

  1. Strong (also matrix, simultaneous, or full) coupling - where the matrix equation is of the form:

    (10–1)

    and the coupled effect is accounted for by the presence of the off-diagonal submatrices [K12] and [K21]. This method provides for a coupled response in the solution after one iteration.

  2. Weak (also load vector or sequential) coupling - where the coupling in the matrix equation is shown in the most general form:

    (10–2)

    and the coupled effect is accounted for in the dependency of [K11] and {F1} on {X2} as well as [K22] and {F2} on {X1}. At least two iterations are required to achieve a coupled response.

The following is a list of the types of coupled-field analyses including methods of coupling present in each:

Table 10.2: Coupling Methods

Analysis Category Coupling Method Used Example Applications
Thermal-Structural Analysis S, WHigh temperature turbine
Electromagnetic Analysis SCurrent fed massive conductors
Stranded Coil Analysis SStranded coils
Structural-Magnetic Analysis S, WSolenoids, high energy magnets (MRI), magnetic actuators
Structural-Electromagnetic Analysis S
Structural-Stranded Coil Analysis S
Electro-Thermo-Structural Analysis WElectro-thermal MEMS actuators
Electro-Magneto-Thermo-Structural Analysis WDirect current electromechanical devices in general
Thermal-Electromagnetic Analysis
Piezoelectric Analysis STransducers, resonators
Electrostatic-Structural Analysis S, WDielectric elastomers, air regions in microelectromechanical systems (MEMS)
Thermo-Piezoelectric Analysis S, WSensors and actuators for smart structures
Piezoresistive Analysis WPressure and force sensors
Thermo-Pressure Analysis S, WPiping networks
Acoustic-Structural Analysis S, WAcoustics
Thermo-Electric Analysis S, WHigh temperature electronics, Peltier coolers, thermoelectric generators
Thermal-Magnetic Analysis WDirect current transients: power interrupts, surge protection
Circuit-Magnetic Analysis SCircuit-fed solenoids, transformers, and motors
Structural-Diffusion Analysis S, WHygroscopic swelling of polymers in electronics packaging, oxygen or hydrogen migration in metals
Thermal-Diffusion Analysis S, WTemperature-dependent moisture migration, thermomigration in metallic interconnects
Structural-Thermal-Diffusion Analysis S, WSodium expansion in aluminum reduction cells
Electric-DiffusionS, WElectromigration in PCB interconnects
Thermal-Electric-DiffusionS, WThermomigration and electromigration in PCB interconnects
Structural-Electric-DiffusionS, WHydrostatic stress-migration and electromigration in PCB interconnects
Structural-Thermal-Electric-DiffusionS, WHydrostatic stress-migration, thermomigration, and electromigration in PCB interconnects

where:

S = strong coupling
W = weak coupling

The solution sequence follows the standard finite element methodology. Convergence is achieved when changes in all unknowns (i.e. DOF) and knowns, regardless of units, are less than the values specified (on the CNVTOL command). Some of the coupling described above is always or usually one-way. For example, in Category A, the temperatures affect the displacements of the structure by way of the thermal strains, but the displacements usually do not affect the temperatures.

The following descriptions of coupled phenomena will include:

  1. Applicable element types

  2. Basic matrix equation indicating coupling terms in bold print. In addition to the terms indicated in bold print, any equation with temperature as a degree of freedom can have temperature-dependency in all terms.

  3. Applicable analysis types, including the matrix and/or vector terms possible in each analysis type.

The nomenclature used on the following pages is given in Table 10.3: Nomenclature of Coefficient Matrices at the end of the section. In some cases, element KEYOPTS are used to select the DOF of the element. DOF will not be fully active unless the appropriate material properties are specified. Some of the elements listed may not be applicable for a particular use as it may be only 1D, whereas a 3D element is needed (e.g. FLUID116).

 

10.1.2.1. Thermal-Structural Analysis

(see Derivation of Structural Matrices, Derivation of Heat Flow Matrices, and Thermoelasticity)

  1. Element type: SOLID5, PLANE13, SOLID98, PLANE222, PLANE223, SOLID225, SOLID226, SOLID227, and LINK228

  2. Matrix equation

    1. Strong coupling:

      (10–3)

    2. Weak coupling:

      (10–4)

    where:

    [Kt] = [Ktb] + [Ktc]
    {F} = {Fnd} + {Fpr} + {Fac}
    {Q} = {Qnd} + {Qg} + {Qc}

  3. Analysis types:

    1. Strong coupling: static, transient, or harmonic

    2. Weak coupling: static or transient

Note:  Strong coupling is supported only by PLANE222, PLANE223, SOLID225, SOLID226, and SOLID227.

{Qted} is applicable only to PLANE222, PLANE223, SOLID225, SOLID226, and SOLID227.

 

10.1.2.2. Electromagnetic Analysis

(see Derivation of Electromagnetic Matrices)

  1. Element type: PLANE13, PLANE233, SOLID236, and SOLID237

  2. Matrix equation

    1. Time-integrated electric potential formulation:

      (10–5)

      where:

      {I} = {Ind}

      The above formulation is used with PLANE13. It is also available with KEYOPT(2) = 2 of PLANE233, SOLID236, and SOLID237.

    2. Electric potential formulation:

      (10–6)

      The above formulation is the default for PLANE233, SOLID236, and SOLID237.

  3. Analysis types: Static, harmonic or transient

 

10.1.2.3. Stranded Coil Analysis

(see Stranded Coil Analysis)

  1. Element type: PLANE233, SOLID236, and SOLID237

  2. Matrix equation:

    A-VOLT-EMF formulation

    (10–7)

    where:

    {I} = {Ind}

    The above formulation is used with PLANE233, SOLID236, and SOLID237.

  3. Analysis types: Static, harmonic or transient

 

10.1.2.4. Structural-Magnetic Analysis

(see Derivation of Electromagnetic Matrices and Magnetoelasticity)

  1. Element type: SOLID5, PLANE13, SOLID98, PLANE223, SOLID226, and SOLID227

  2. Matrix equations:

    1. Weak coupling with the scalar potential formulation using SOLID5 and SOLID98:

      (10–8)

    2. Weak coupling with the magnetic vector potential formulation using PLANE13:

      (10–9)

    3. Strong coupling with the magnetic vector potential formulation using PLANE223 and the edge-flux formulation using SOLID226 and SOLID227.

      1. Maxwell force method (KEYOPT(8) = 0):

        (10–10)

      2. Lorentz force method (KEYOPT(8) = 1):

        (10–11)

    where:

    {F} = {Fnd} + {Fpr} + {Fac} +{Fth}

  3. Analysis types: static or transient with PLANE13, PLANE223, SOLID226, and SOLID227 elements; static analysis only with SOLID5 and SOLID98

 

10.1.2.5. Structural-Electromagnetic Analysis

(see Derivation of Electromagnetic Matrices and Magnetoelasticity)

  1. Element type: PLANE223, SOLID226, and SOLID227

  2. Matrix equation:

    1. Strong coupling with the Maxwell force option (KEYOPT(8) = 0):

      (10–12)

    2. Strong coupling with the Lorentz force option (KEYOPT(8) = 1):

      (10–13)

    where:

    {F} = {Fnd} + {Fpr} + {Fac} +{Fth}

  3. Analysis types: static or transient

 

10.1.2.6. Structural-Stranded Coil Analysis

(see Derivation of Electromagnetic Matrices and Magnetoelasticity)

  1. Element type: PLANE223, SOLID226, and SOLID227

  2. Matrix equation:

    1. Strong coupling with the Maxwell force option (KEYOPT(8) = 0):

      (10–14)

    2. Strong coupling with the Lorentz force option (KEYOPT(8) = 1):

      (10–15)

    where:

    {F} = {Fnd} + {Fpr} + {Fac} +{Fth}

  3. Analysis types: static or transient

 

10.1.2.7. Electro-Thermo-Structural Analysis

(see Derivation of Structural Matrices, Derivation of Heat Flow Matrices, Thermoelasticity, and Thermoelectrics)

  1. Element type: PLANE222, PLANE223, SOLID225, SOLID226, SOLID227, and LINK228

  2. Matrix equation:

    (10–16)

    where:

    [Kt] = [Ktb] + [Ktc]
    {F} = {Fnd} + {Fpr} + {Fac}
    {Q} = {Qnd} + {Qg} + {Qc} + {Qj} + {Qp}
    {I} = {Ind}

  3. Analysis types: static and transient

 

10.1.2.8. Electro-Magneto-Thermo-Structural Analysis

(see Derivation of Electromagnetic Matrices and Piezoelectrics)

  1. Element types: SOLID5 and SOLID98

  2. Matrix equation:

    (10–17)

    where:

    [Kt] = [Ktb] + [Ktc]
    {F} = {Fnd} + {Fth} + {Fac} + {Fjb} + {Fpr} + {Fmx}
    {Q} = {Qnd} + {Qg} + {Qj} + {Qc}
    {I} = {Ind}

  3. Analysis types: Static or Transient

 

10.1.2.9. Thermal-Electromagnetic Analysis

(see Derivation of Electromagnetic Matrices and Derivation of Heat Flow Matrices)

  1. Element types: SOLID5, SOLID98, PLANE223, SOLID226, and SOLID227

  2. Matrix equation

    1. Magnetic scalar potential formulation for SOLID5, SOLID98:

      (10–18)

    2. Magnetic vector potential formulation for PLANE223:

      (10–19)

      Note:  Electric and magnetic vector potentials are strongly coupled.

      In this formulation, the VOLT degree of freedom is the actual electric potential (not the time integrated electric potential as given in Equation 5–92). Therefore, matrices and appear in the stiffness matrix in this unsymmetric strong coupling form.

    3. Edge-flux formulation for SOLID226 and SOLID227:

      (10–20)

    where:

    [Kt] = [Ktb] + [Ktc]
    {Q} = {Qnd} + {Qg} + {Qj} + {Qc}
    {I} = {Ind}

  3. Analysis types: Static or Transient

 

10.1.2.10. Piezoelectric Analysis

(see Piezoelectrics)

  1. Element types: SOLID5, PLANE13, SOLID98, PLANE222, PLANE223, SOLID225, SOLID226, and SOLID227.

  2. Matrix equation:

    (10–21)

    where:

    {F} = {Fnd} + {Fth} + {Fac} + {Fpr}
    {L} = {Lnd} + {Lc} + {Lsc}+{Lth}

    Note:  {Lc} and {Lsc} are applicable to only PLANE222, PLANE223, SOLID225, SOLID226, and SOLID227.

  3. Analysis types: Static, modal, harmonic, or transient

 

10.1.2.11. Electrostatic-Structural Analysis

(see Electroelasticity)

  1. Element types: PLANE222, PLANE223, SOLID225, SOLID226, and SOLID227.

  2. Matrix equation

    1. Strong coupling:

      (10–22)

    2. Weak coupling:

      (10–23)

    where:

    {F} = {Fnd} + {Fth} + {Fac} + {Fpr} + {Fe}
    {L} = {Lnd} + {Lc} + {Lsc}

  3. Analysis types:

    1. Strong Coupling: nonlinear static or transient; linear perturbation static, modal, or harmonic

    2. Weak coupling: static or transient

 

10.1.2.12. Thermo-Piezoelectric Analysis

(see Derivation of Structural Matrices, Derivation of Heat Flow Matrices, Thermoelasticity, and Piezoelectrics)

  1. Element type: PLANE222, PLANE223, SOLID225, SOLID226, and SOLID227

  2. Matrix equation

    1. Strong coupling:

      (10–24)

    2. Weak coupling:

      (10–25)

    where:

    [Kt] = [Ktb] + [Ktc]
    {F} = {Fnd} + {Fpr} + {Fac}
    {Q} = {Qnd} + {Qg} + {Qc}
    {L} = {Lnd} + {Lc} + {Lsc}

  3. Analysis types:

    1. Strong coupling: static, transient, harmonic, modal

    2. Weak coupling: static or transient

 

10.1.2.13. Piezoresistive Analysis

(see Piezoresistivity)

  1. Element type: PLANE222, PLANE223, SOLID225, SOLID226, and SOLID227

  2. Matrix equation:

    (10–26)

    where:

    [Kv] = conductivity matrix (see Equation 10–107) updated for piezoresistive effects
    {F} = {Fnd} + {Fth} + {Fpr} + {Fac)
    {I} = {Ind}

  3. Analysis types: Static or transient

 

10.1.2.14. Thermo-Pressure Analysis

(see FLUID116 - Coupled Thermal-Fluid Pipe)

  1. Element type: FLUID116

  2. Matrix equation:

    (10–27)

    where:

    [Kt] = [Ktb] + [Ktc] + [Ktm]
    {Q} = {Qnd} + {Qc} + {Qg}
    {W} = {Wnd} + {Wh}

  3. Analysis types: Static or Transient

 

10.1.2.15. Acoustic-Structural Analysis

(See Derivation of Acoustic Matrices.)

  1. Element type: FLUID29, FLUID30, FLUID220, and FLUID221 (with other structural elements)

  2. Matrix equation:

    (10–28)

    Values for [M], [C], and [K] are provided by other elements.

  3. Analysis types: Transient, harmonic and modal analyses can be performed.

 

10.1.2.16. Thermo-Electric Analysis

  1. Element types: SOLID5, LINK68, SOLID98, SHELL157, PLANE222, PLANE223, SOLID225, SOLID226, SOLID227, and LINK228

  2. Matrix equation:

    (10–29)

    where:

    [Kt] = [Ktb] + [Ktc]
    {Q} = {Qnd} + {Qc} + {Qg} + {Qj} + {Qp}
    {I} = {Ind}

    Note:  {Qp}, [Kvt], and [Cv] are used only for PLANE222, PLANE223, SOLID225, SOLID226, and SOLID227.

  3. Analysis types: Static or Transient

 

10.1.2.17. Thermal-Magnetic Analysis

(see Derivation of Electromagnetic Matrices and Derivation of Heat Flow Matrices)

  1. Element types: PLANE13, PLANE223, SOLID226, and SOLID227

  2. Matrix equation:

    (10–30)

    where:

    [Kt] = [Ktb] + [Ktc]
    {Q} = {Qnd} + {Qg} + {Qj} + {Qc}

  3. Analysis types: Static or Transient

 

10.1.2.18. Circuit-Magnetic Analysis

(See Stranded Coil Analyses)

  1. Element type: CIRCU124

  2. Matrix equation:

    (10–31)

  3. Analysis types: Static, Transient, or Harmonic

 

10.1.2.19. Structural-Diffusion Analysis

(see Derivation of Structural Matrices, Derivation of Diffusion Matrices, and Structural-Diffusion Coupling)

  1. Element type: PLANE222, PLANE223, SOLID225, SOLID226, and SOLID227

  2. Matrix equation

    1. Strong coupling:

      (10–32)

    2. Weak coupling:

      (10–33)

      where:

      {F} = {Fnd}+ {Fpr}+ {Fac}

  3. Analysis types: static and transient

 

10.1.2.20. Thermal-Diffusion Analysis

(see Derivation of Heat Flow Matrices, Derivation of Diffusion Matrices)

  1. Element type: PLANE222, PLANE223, SOLID225, SOLID226, and SOLID227

  2. Matrix equation

    (10–34)

    where:

    [Kt] = [Ktb] + [Ktc]

    {Q} = {Qnd}+ {Qc}+ {Qg}+ {Qp}

  3. Analysis types: static and transient

 

10.1.2.21. Structural-Thermal-Diffusion Analysis

(see Derivation of Structural Matrices, Derivation of Heat Flow Matrices, Derivation of Diffusion Matrices, Thermoelasticity, and Structural-Diffusion Coupling)

  1. Element type: PLANE222, PLANE223, SOLID225, SOLID226, and SOLID227

  2. Matrix equation

    1. Strong coupling:

      (10–35)

    2. Weak coupling:

      (10–36)

    where:

    {F} = {Fnd}+ {Fpr}+ {Fac}

    {Q} = {Qnd}+ {Qc}+ {Qg}+ {Qp}

  3. Analysis type: static and transient

Table 10.3: Nomenclature of Coefficient Matrices

SymbolMeaningUsage
[M]structural mass matrix (discussed in Derivation of Structural Matrices)[1]
[Mfs]fluid-structure coupling mass matrix (discussed in Derivation of Acoustic Matrices)[1]
[Mp]acoustic mass matrix (discussed in Derivation of Acoustic Matrices)[1]
[C]structural damping matrix (discussed in Derivation of Structural Matrices)[2]
[Ct]thermal specific heat matrix (discussed in Derivation of Heat Flow Matrices)[2]
[Ctu]thermoelastic damping matrix (discussed in Thermoelasticity)[2]
[CAA]magnetic damping matrix (discussed in Electromagnetic Field Evaluations)[2]
[Cp]acoustic damping matrix (discussed in Derivation of Acoustic Matrices)[2]
[CAv]magnetic-electric damping matrix (discussed in Derivation of Electromagnetic Matrices)[2]
[Cvv]electric damping matrix (discussed in Derivation of Electromagnetic Matrices)[2]
[CiA]inductive damping matrix (discussed in Stranded Coil Analyses)[2]
[Cv]dielectric permittivity coefficient matrix (discussed in Quasistatic Electric Analysis)[2]
[Cvh]dielectric damping matrix (discussed in Quasistatic Electric Analysis)[2]
[K]structural stiffness matrix (discussed in Derivation of Structural Matrices)[3]
[Kt]thermal conductivity matrix (may consist of 1, 2, or 3 of the following 3 matrices) (discussed in Derivation of Heat Flow Matrices)[3]
[Ktb]thermal conductivity matrix of material (discussed in Derivation of Heat Flow Matrices)[3]
[Ktc]thermal conductivity matrix of convection surface (discussed in Derivation of Heat Flow Matrices)[3]
[Ktm]thermal conductivity matrix associated with mass transport (discussed in Derivation of Heat Flow Matrices)[3]
[Kut]thermoelastic stiffness matrix (discussed in Thermoelasticity)[3]
[Km]scalar magnetic potential coefficient matrix (discussed in Derivation of Electromagnetic Matrices)[3]
[KAA]magnetic reluctivity matrix (discussed in Derivation of Electromagnetic Matrices)[3]
[KVV]electric conductivity matrix for electromagnetic analysis (discussed in Derivation of Electromagnetic Matrices)[3]
[KAV]magneto-electric coupling matrix (discussed in Derivation of Electromagnetic Matrices)[3]
[KVA]electromagnetic coupling matrix (discussed in Derivation of Electromagnetic Matrices)[3]
[KAi]potential-current coupling stiffness matrix (discussed in Stranded Coil Analyses)[3]
[Kii]resistive stiffness matrix (discussed in Stranded Coil Analyses)[3]
[Kie]current-emf coupling stiffness (discussed in Stranded Coil Analyses)[3]
[Kv]electrical conductivity coefficient matrix for electric analysis (discussed in Derivation of Electromagnetic Matrices)[3]
[Kz]piezoelectric stiffness matrix (discussed in Piezoelectrics)[3]
[Keu]electric force stiffness or electrostatic softening matrix (discussed in Electroelasticity)[3]
[KeV]electric force coupling matrix (discussed in Electroelasticity)[3]
[Kzt]thermo-piezoelectric stiffness matrix (discussed in Piezoelectrics)[3]
[Kd]dielectric coefficient matrix (discussed in Piezoelectrics)[3]
[Kvt]Seebeck coefficient coupling matrix[3]
[Kud]Diffision-elastic stiffness matrix (discussed in Structural-Diffusion Coupling)[3]
Maxwell stress-stiffening matrix (discussed in Magnetoelasticity)[3]
Maxwell stress coupling matrix (discussed in Magnetoelasticity)[3]
Lorentz force coupling matrix (discussed in Magnetoelasticity)[3]

  1. Coefficient matrices of second time derivatives of unknowns.

  2. Coefficient matrices of first time derivative of unknowns

  3. Coefficient matrices of unknowns

Vectors of Knowns

SymbolMeaningAssociated Input / Output Label
{Fnd}applied nodal force vector (discussed in Derivation of Structural Matrices)FX ... MZ
{Fnr}Newton-Raphson restoring load vector (discussed in Newton-Raphson Procedure FX ... MZ
{Fth}thermal strain force vector (discussed in Derivation of Structural Matrices)FX ... MZ
{Fpr}pressure load vector (discussed in Derivation of Structural Matrices)FX ... MZ
{Fac}force vector due to acceleration effects (that is, gravity) (discussed in Derivation of Structural Matrices)FX ... MZ
{Fjb}Lorentz force vector (discussed in Derivation of Electromagnetic Matrices)FX ... FZ
{Fmx}Maxwell force vector (discussed in Derivation of Electromagnetic Matrices)FX ... FZ
{Fe}electrostatic body-force load vector (discussed in Electroelasticity)FX ...FZ
{Fb}body-force load vector due to non-gravity effects (discussed in Derivation of Heat Flow Matrices)FX ... MZ
{Fdi}

diffusion strain force vector (discussed in Structural-Diffusion Coupling)

FX ... MZ
{Qnd}applied nodal heat flow rate vector (discussed in Derivation of Heat Flow Matrices)HEAT, HBOT, HE2, ... HTOP
{Qf}heat flux vector (discussed in Derivation of Heat Flow Matrices)HEAT, HBOT, HE2, ... HTOP
{Qc}convection surface vector (discussed in Derivation of Heat Flow Matrices)HEAT, HBOT, HE2, ... HTOP
{Qg}heat generation rate vector for causes other than Joule heating (discussed in Derivation of Heat Flow Matrices)HEAT, HBOT, HE2, ... HTOP
{Qj}heat generation rate vector for Joule heating (discussed in Electromagnetic Field Evaluations)HEAT
{Qp}Peltier heat flux vector (discussed in Thermoelectrics)HEAT
{Qted}heat generation rate vector for thermoelastic dampingHEAT
applied nodal source current vector (associated with {A}) (discussed in Derivation of Electromagnetic Matrices)CSGZ
applied nodal flux vector (associated with {ϕ}) (discussed in Derivation of Electromagnetic Matrices)FLUX
source (Biot-Savart) vector (discussed in Derivation of Electromagnetic Matrices)FLUX
coercive force (permanent magnet) vector (discussed in Derivation of Electromagnetic Matrices)FLUX
source current vector (discussed in Derivation of Electromagnetic Matrices)FLUX
{Ind}applied nodal electric current vector (discussed in Derivation of Electromagnetic Matrices)AMPS
{Lnd}applied nodal charge vector (discussed in Piezoelectrics)AMPS (CHRG for PLANE222, PLANE223, SOLID225, SOLID226, and SOLID227)
{Lc}charge density load vector (discussed in Derivation of Electromagnetic Matrices)CHRGD
{Lsc}surface charge density load vector (discussed in Derivation of Electromagnetic Matrices)CHRGS
{Lth}thermo-piezoelectric load vector (discussed in Piezoelectrics)TEMP, EPTH
{Wnd}applied nodal fluid flow vector (discussed in FLUID116 - Coupled Thermal-Fluid Pipe)FLOW
{Wh}static head vector (discussed in FLUID116 - Coupled Thermal-Fluid Pipe)FLOW
{R}diffusion flow rate vectorRATE

Vectors of Unknowns

{u}displacement vector (discussed in Derivation of Structural Matrices)UX ... ROTZ
{T}thermal potential (temperature) vector (discussed in Derivation of Heat Flow Matrices)TEMP, TBOT, TE2, ... TTOP
{V}electric potential vector (discussed in Derivation of Electromagnetic Matrices)VOLT
{ΔV}voltage drop in a stranded coil analysis (discussed in Stranded Coil Analysis)VOLT
{ν}time integrated electric potential vector (discussed in Derivation of Electromagnetic Matrices)VOLT
{φ}magnetic scalar potential vector (discussed in Derivation of Electromagnetic Matrices)MAG
{A}magnetic vector potential or edge-flux (discussed in Derivation of Electromagnetic Matrices)

AZ

{i}electric current vector (discussed in Stranded Coil Analyses)CURR
{e}electromagnetic force drop vector (discussed in Stranded Coil Analyses)EMF
{P}pressure vector (discussed in and Derivation of Acoustic Matrices)PRES
{C}concentration vector (discussed in Derivation of Diffusion Matrices)CONC
. time derivative 
. . second time derivative