A new adaptive multiscale method based on the estimate of residual forces for static analysis of heterogeneous materials

• A new adaptive multiscale method (AMM) is developed.
• The estimate of residual force is taken as local relative error indicator.
• The validity of the proposed algorithm has been verified.
• Nearly optimal distribution of macroscopic nodes can be found by using AMM.

A new adaptive multiscale method (AMM) is developed based on the estimate of residual forces for static analysis of heterogeneous materials. The AMM is established by combining multi-node extended multiscale finite element method (multi-node EMsFEM) with a new proposed macroscopic node adaptive algorithm. In our previous multiscale computations, macroscopic nodes are placed uniformly along each edge of multi-node coarse element without considering local strain or displacement gradient. In this paper, to optimize the distribution of macroscopic nodes, a new adaptive algorithm is proposed based on the estimate of residual forces. Numerical experiments have indicated that residual forces exist even for linear elastic problems. For boundary external loading cases, residual forces only exist on the edges of coarse element. Besides, computations indicate that residual forces can reflect local relative errors in the multi-node EMsFEM computations. Thus it is reasonable and suitable to take residual forces as local relative error indicators in the multi-node EMsFEM computations. Finally, the AMM is developed based on this idea. To verify the validity of this proposed method, three typical numerical examples are carried out. The examples demonstrate that nearly optimal distributions of macroscopic nodes can be obtained by employing the proposed AMM.