Formulating finite elements representing a given microstructure without using homogenisation

A framework is proposed to directly calculate the stiffness matrix of a macro-element equivalent to a desired microstructure. In a sense, this approach enables the estimation of the “homogenised” properties of microstructures without using the homogenisation theory. The proposed framework is based on assumed displacement fields within equivalent macro-elements. Different displacement assumptions are briefly discussed and two equivalent macro-elements are formulated. Several numerical tests are performed and discussed to show the potentials and limitations of the equivalent macro-elements. The numerical results show that one of the formulated elements can provide reasonable accuracy improvements, particularly in stress calculations. A sensitivity analysis is provided allowing the proposed equivalent macro-elements to be used in topology optimisation. The numerical results show that the formulated elements can effectively enhance the resolution of the boundaries in optimised solutions to obtain smooth designs.