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

Abstract

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 problems using statistical estimation methodology

“…This section presents the linear SDM, previous nonlinear SDM, and new SDM. We consider stationary heat conduction in a 1‐dimensional heterogeneous rod as an example.…”

“…As we repeat the local analysis m l times in the previous SDM, the computational cost taken in constructing N i and a i is m l times that of the linear SDM. Although increasing m l would enhance the calculation accuracy, the nonlinear technique cannot accurately solve a problem with a strong nonlinearity even if m l is considerably large . One reason for this is that the microscopic regions of influence are analyzed employing a linear FEM without a nonlinear solver.…”

“…To investigate the calculation precision and computational cost, we solved a nonlinear problem of 2‐dimensional stationary heat conduction employing 3 methods: the direct nonlinear FEM with 4‐node quadrangular elements using general‐purpose analysis software ANSYS, the proposed nonlinear SDM with 8‐CP regions of influence ( m = 8), the previous nonlinear SDM (see Suzuki et al and Sections 2.2 and 3.2.2) with 8‐CP regions of influence ( m = 8), and the homogenized nonlinear FEM with 4‐node quadrangular elements. …”

“…Previous work presented a multiscale numerical solver called the seamless‐domain method (SDM). The SDM model is meshless and represented by only a small number of coarse‐grained points (CPs).…”

“…Initially, the SDM framework was constructed for solving linear problems. However, we developed a rapid and low‐cost nonlinear solver for the SDM technique . Like the linear SDM, the nonlinear solver does not require iterative multiscale analysis for every convergence calculation, which greatly reduces the computational cost.…”