17.6. POST1 - Error Approximation Technique

17.6.1. Error Approximation Technique for Displacement-Based Problems

The error approximation technique used by POST1 (PRERR command) for displacement-based problems is similar to that given by Zienkiewicz and Zhu([102]). The essentials of the method are summarized below.

The usual continuity assumption used in many displacement based finite element formulations results in a continuous displacement field from element to element, but a discontinuous stress field. To obtain more acceptable stresses, averaging of the element nodal stresses is done. Then, returning to the element level, the stresses at each node of the element are processed to yield:

(17–80)

where:

= stress error vector at node n of element i

= number of elements connecting to node n

= stress vector of node n of element i

Then, for each element

(17–81)

where:

e_i = energy error for element i (accessed with ETABLE (SERR item) command)

vol = volume of the element (accessed with ETABLE (VOLU item) command)

[D] = stress-strain matrix evaluated at reference temperature

{Δσ} = stress error vector at points as needed (evaluated from all {Δσ_n} of this element)

The energy error over the model is:

(17–82)

where:

e = energy error over the entire (or part of the) model (accessed with *GET (SERSM item) command)

N_r = number of elements in model or part of model

The energy error can be normalized against the strain energy.

(17–83)

where:

E = percentage error in energy norm (accessed with PRERR, PLDISP, PLNSOL (U item), *GET (SEPC item) commands)

U = strain energy over the entire (or part of the) model (accessed with *GET (SENSM item) command)

= strain energy of element i (accessed with ETABLE (SENE item) command) (see Energies)

The e_i values can be used for adaptive mesh refinement. It has been shown by Babuska and Rheinboldt([103]) that if e_i is equal for all elements, then the model using the given number of elements is the most efficient one. This concept is also referred to as “error equilibration”.

At the bottom of all printed nodal stresses (the PRNSOL or PRESOL command), which may consist of the 6 component stresses, the 5 combined stresses, or both, a summary printout labeled: ESTIMATED BOUNDS CONSIDERING THE EFFECT OF DISCRETIZATION ERROR gives minimum nodal values and maximum nodal values. These are:

(17–84)

(17–85)

where min and max are over the selected nodes, and

where:

= nodal minimum of stress quantity (output as VALUE (printout) or SMNB (plot))

= nodal maximum of stress quantity (output as VALUE (printout) or SMXB (plot) )

j = subscript to refer to either a particular stress component or a particular combined stress

= average of stress quantity j at node n of element attached to node n

= maximum of stress quantity j at node n of element attached to node n

Δσ_n = root mean square of all Δσ_i from elements connecting to node n

Δσ_i = maximum absolute value of any component of

for all nodes connecting to element (accessed with ETABLE (SDSG item) command)

17.6.2. Error Approximation Technique for Temperature-Based Problems

The error approximation technique used by POST1 (PRERR command) for temperature based problems is similar to that given by Huang and Lewis([126]). The essentials of the method are summarized below.

The usual continuity assumption results in a continuous temperature field from element to element, but a discontinuous thermal flux field. To obtain more acceptable fluxes, averaging of the element nodal thermal fluxes is done. Then, returning to the element level, the thermal fluxes at each node of the element are processed to yield:

(17–86)

where:

= thermal flux error vector at node n of element i

= number of elements connecting to node n

= thermal flux vector of node n of element

Then, for each element

(17–87)

where:

e_i = energy error for element i (accessed with ETABLE (TERR item) command)

vol = volume of the element (accessed with ETABLE (VOLU item) command)

[D] = conductivity matrix evaluated at reference temperature

{Δq} = thermal flux error vector at points as needed (evaluated from all {Δq_n} of this element)

The energy error over the model is:

(17–88)

where:

e = energy error over the entire (or part of the) model (accessed with *GET (TERSM item) command)

N_r = number of elements in model or part of model

The energy error can be normalized against the thermal dissipation energy.

(17–89)

where:

E = percentage error in energy norm (accessed with PRERR, PLNSOL, (TEMP item) or *GET (TEPC item) commands)

U = thermal dissipation energy over the entire (or part of the) model (accessed with *GET (TENSM item) command)

= thermal dissipation energy of element i (accessed with ETABLE (TENE item) command) (see Energies)

At the bottom of all printed fluxes (with the PRNSOL command), which consists of the 3 thermal fluxes, a summary printout labeled: ESTIMATED BOUNDS CONSIDERING THE EFFECT OF DISCRETIZATION ERROR gives minimum nodal values and maximum nodal values. These are:

(17–90)

(17–91)

where min and max are over the selected nodes, and

where:

= nodal minimum of thermal flux quantity (output as VALUE (printout) or SMNB (plot))

= nodal maximum of thermal flux quantity (output as VALUE (printout) or SMXB (plot))

j = subscript to refer to either a particular thermal flux component or a particular combined thermal flux

= average of thermal flux quantity j at node n of element attached to node n

= maximum of thermal flux quantity j at node n of element attached to node n

Δq_n = maximum of all Δq_i from elements connecting to node n

Δq_i = maximum absolute value of any component of

for all nodes connecting to element (accessed with ETABLE (TDSG item) command)

17.6.3. Error Approximation Technique for Magnetics-Based Problems

The error approximation technique used by POST1 (PRERR command) for magnetics- based problems is similar to that given by Zienkiewicz and Zhu ([102]) and Huang and Lewis ([126]). The essentials of the method are summarized below.

The usual continuity assumption results in a continuous temperature field from element to element, but a discontinuous magnetic flux field. To obtain more acceptable fluxes, averaging of the element nodal magnetic fluxes is done. Then, returning to the element level, the magnetic fluxes at each node of the element are processed to yield:

(17–92)

where:

= magnetic flux error vector at node n of element i

= number of elements connecting to node n

= magnetic flux vector of node n of element

Then, for each element

(17–93)

where:

e_i = energy error for element i (accessed with ETABLE (BERR item) command)

vol = volume of the element (accessed with ETABLE (VOLU item) command)

[D] = magnetic conductivity matrix evaluated at reference temperature

{ΔB} = magnetic flux error vector at points as needed (evaluated from all {ΔB_n} of this element)

The energy error over the model is:

(17–94)

where:

e = energy error over the entire (or part of the) model (accessed with *GET (BERSM item) command)

N_r = number of elements in model or part of model

The energy error can be normalized against the magnetic energy.

(17–95)

where:

E = percentage error in energy norm (accessed with PRERR, PLNSOL, (TEMP item) or *GET (BEPC item) commands)

U = magnetic energy over the entire (or part of the) model (accessed with *GET (BENSM item) command)

= magnetic energy of element i (accessed with ETABLE (SENE item) command) (see Energies)

At the bottom of all printed fluxes (with the PRNSOL command), which consists of the 3 magnetic fluxes, a summary printout labeled: ESTIMATED BOUNDS CONSIDERING THE EFFECT OF DISCRETIZATION ERROR gives minimum nodal values and maximum nodal values. These are: