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In this paper, we applied a multiscale numerical scheme called the seamless-domain method (SDM) to nonlinear elliptic boundary value problems. Although the SDM is meshfree, it can obtain a high-resolution solution whose dependent-variable gradient(s) is sufficiently smooth and continuous. The SDM models with only coarse-grained points can produce accurate solutions for both linear heat conduction problems and linear elastic problems. This manuscript presents a simple nonlinear solver for the SDM analysis of heterogeneous materials. Although the solver can easily approximate the solutions to nonlinear multiscale problems, it does not require an iterative multiscale analysis at every convergence calculation. In other words, the proposed scheme does not completely interactively couple the multiple scales. We present numerical examples of nonlinear stationary heat conduction analyses of heterogeneous fields and compare the SDM model, the direct finite-element model, and the homogenized model based on the homogenization theory. For a real heterogeneous structure (graphite fiber composite) that did not have strong material nonlinearities, the SDM model using only 925 points gave a solution with similar precisions as an ordinary finite element solution using hundreds of thousands of nodes. To investigate the limitations of the method, we also applied the SDM to imaginary materials with various strengths of thermal property nonlinearities.

Multiscale seamless-domain method for solving nonlinear heat conduction problems without iterative multiscale calculations