On the efficient enforcement of uniform traction and mortar periodic boundary conditions in computational homogenisation

In this contribution, the enforcement of uniform traction (UT) and periodic boundary (PB) conditions on arbitrarily generated meshes is analysed in detail, and the derivation of the associated macroscopic consistent tangent operators is described. Two different approaches to impose these boundary conditions are examined: the saddle point solution (SPS) resulting from the Lagrange multiplier method and the condensation method (CM). The numerical treatment required to implement the UT and the PB conditions with both methods is presented for rectangular and cuboidal representative volume elements (RVEs). It is shown that the UT condition only needs rigid body motion to be prescribed since it intrinsically restricts rigid body rotations in the finite strain setting. The application of the SPS to the UT condition leads to a formulation where the Lagrange multipliers are the homogenised stresses, and the macroscopic consistent tangent operator for FE2 simulations can be directly derived. The enforcement of PB conditions on non-conforming meshes with the mortar approach is also described for the CM and SPS. The need to modify the Lagrange multiplier dual interpolation functions is emphasised to avoid over-constraints. The proposed implementation recovers the conventional PB condition solution when conforming meshes are used. The computational performance of both methods is compared in terms of time and memory requirements. As anticipated, the results obtained for each boundary condition do not depend on the enforcement method employed. With the direct solvers employed, the SPS is more efficient despite increasing the number of unknowns in the linear system of equations. •A comparison of two enforcement strategies for micro-scale constraints is undertaken.•An efficient strategy to determine the homogenised consistent tangent is devised.•Dual mortar basis functions are modified to avoid over-constraints.•Assessment of the mortar periodic boundary condition accuracy is performed.•The Saddle Point approach is more efficient than the Condensation Method.