Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework

A convergence study is presented for a form of gradient elasticity where the enrichment is through the Laplacian of the strain, so that a fourth-order partial differential equation results. Isogeometric finite element analysis is used to accommodate the higher continuity required by the inclusion of strain gradients. A convergence analysis is carried out for the original system of a fourth-order partial differential equation. Both global refinement, using NURBS, and local refinement, using T-splines, have been applied. Theoretical convergence rates are recovered, except for a polynomial order of two, when the convergence rate is suboptimal, a result which also has been found for the (fourth-order) Cahn-Hilliard equation. The convergence analyses have been repeated for the case that an operator split is applied so that a set of two (one-way) coupled partial differential equations results. Differences occur with the results obtained for the original fourth-order equation, which is caused by the boundary conditions, which is the first time this effect has been substantiated.