Error Estimation

There are two mesh refinement strategies that are implemented in TMR: the first is based on the strain energy norm, while the second is an adjoint-based refinement technique. The strain energy based refinement method is designed to reduce the solution error in the natural strain energy norm. This call takes the form:

tmr.TMR.strainEnergyError()
strainEnergyError(forest, coarse_assembler, forest_refined,

refined_assembler)

The following function performs a mesh refinement based on a strain energy criteria. It is based on the following relationship for linear finite-element analysis

\[\Pi(u-u_h,u-u_h) = \Pi(u,u) - \Pi(u_h,u_h)\]

where a(u,u) is the trilinear strain energy functional, u is the exact solution, and uh is the discretized solution at any mesh level. This relies on the relationship that \(\Pi(u_h, u - u_h) = 0\) which is satisfied due to the method of Galerkin/Ritz.

The following function computes a localized error indicator using the element-wise strain energy. The code computes a higher-order reconstructed solution using a cubic enrichment functions. These enrichment functions expand the original solution space and are computed based on a least-squares approximation with nodal gradient values. The localized error indicator is evaluated as follows:

\[err = \sum_{i=1}^{4} \Pi_e(u_{ce}, u_{ce}) - \Pi_e(u_e, u_e)\]

where \(u_{ce}\) is the element-wise cubic element reconstruction projected onto a uniformly refined mesh.

Parameters

  • forest_coarse (OctForest or QuadForest) – Forest for current mesh level

  • coarse_assembler (Assembler) – Finite assembler class for associated with forest

  • forest_refined (OctForest or QuadForest) – Forest for refined mesh level

  • refined_assembler (Assembler) – Finite assembler class for associated with forest_refined

Returns

  • ans (double) – Total strain energy error

  • err (array of double) – Elemental strain energy error

The forest can be refined based on the requested target error and the solution set in TACS.

The adjoint-based refinement method requires an Assembler object for both the current level of refinement and a uniformly refined version of the model. The adjoint-based mesh refinement takes the form:

tmr.TMR.adjointError()
adjointError(forest, coarse_assembler, forest_refined, refined_assembler,

solution_refined, adjoint_refined)

Refine the mesh using the original solution and the adjoint solution

Parameters

  • forest (OctForest or QuadForest) – Forest for current mesh level

  • coarse_assembler (Assembler) – Finite assembler class for associated with forest

  • forest_refined (OctForest or QuadForest) – Higher-order forest for refined mesh level

  • refined_assembler (Assembler) – Higher-order finite assembler class for associated with forest_refined

  • solution_refined (Vec) – The higher-order solution (or approximation)

  • adjoint_refined (Vec) – The difference between the refined and coarse adjoint solutions computed in some manner

Returns

  • ans (double) – Total strain energy error

  • err (array of double) – Elemental strain energy error

  • adj_corr (TacsScalar) – Adjoint-based functional correction