Field Equations
Field equations can be modeled separately or fully coupled. Some fields in Abaqus/Standard can only have linear response. Each field is discretized by using basic nodal variables (the degrees of freedom at the nodes of the finite element model) such as the components of the displacement in a continuum stress analysis problem. Each field has a conjugate “flux.”
Available Fields and Their Conjugate Fluxes
The fields and conjugate fluxes available in Abaqus/Standard are as follows:
Basic problem | Field | Conjugate flux |
---|---|---|
Stress analysis: force equilibrium | Displacement, u | Force, F |
Stress analysis: moment equilibrium | Rotation, ϕ | Moment, M |
Stress analysis: analysis containing beams with warping | Warping, w | Bimoment, W |
Heat transfer analysis | Temperature, θ | Heat flux, q |
Acoustic analysis (linear only) | Acoustic pressure, u | Rate of change of fluid volumetric flux |
Pore liquid flow analysis | Pore liquid pressure, u | Pore liquid volumetric flux, q |
Hydrostatic fluid modeling | Fluid pressure, p | Fluid volume, V |
Mass diffusion analysis | Normalized concentration, ϕ | Mass concentration volumetric flux, Q |
Piezoelectric analysis | Electrical potential, φ | Electrical charge, q |
Electric conduction analysis | Electrical potential, φ | Electrical current, J |
Mechanism analysis (connector elements with material flow degree of freedom) | Material flow | Material flux |
Analysis containing C3D4H elements (all materials, except compressible hyperelastic elastomers and elastomeric foams). | Pressure Lagrange multiplier | Volumetric flux |
Analysis containing C3D4H elements with compressible hyperelastic or hyperfoam materials. | Volumetric Lagrange multiplier | Pressure flux |
Constraint Equations
In some cases the problem also involves constraint equations. In Abaqus/Standard the following constraints are included by using Lagrange multipliers:
Problem | Constraint variable | Constraint |
---|---|---|
Hybrid solid (except C3D4H elements) | Pressure stress | Volumetric strain compatibility |
Hybrid beam | Axial force | Axial strain compatibility |
Hybrid beam | Transverse shear force | Transverse shear strain compatibility |
Distributing coupling | Force | Coupling displacement compatibility |
Distributing coupling | Moment | Coupling rotation compatibility |
Contact | Normal pressure | Surface penetration |
Contact with Lagrange friction | Shear stress | Relative shear sliding |
If the penalty method is used, the contact Lagrange multipliers may not be present.
Solving Coupled Field Equations
In a general problem several (possibly nonlinear) coupled field equations of types α=1,2,…N must be solved and several different (possibly nonlinear) constraints of type j=1,2,…K must be satisfied simultaneously. For example, in a structural problem in which hybrid beam elements are used, α=1 might represent the displacement field and the equilibrium equations for the conjugate force and α=2 might represent the rotation field and the equilibrium equations for the conjugate moment, while j=1 represents axial strain compatibility and j=2 represents transverse shear strain compatibility.