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Clustering solver for displacement-based Numerical homogenization

Theoretical & Applied Mechanics Letters 2022年第3期12卷 207-212页

作者： Shaoqiang Tang Xi Zhu HEDPS CAPT and LTCSCollege of EngineeringPeking UniversityBeijing 100871China

Based on strain-clustering via k-means,we decompose computational domain into clusters of possibly disjoint cells.Assuming cells in each cluster take the same strain,we reconstruct displacement field.We further propose a new way to condensate the governing equations of displacement-based finite element method to reduce the complexity while maintain the accuracy.numerical examples are presented to illustrate the efficiency of the clustering solver.numerical convergence studies are performed for the examples.Avoiding complexities which is common in existing clustering analysis methods,the proposed clustering solver is easy to implement,particularly for Numerical homogenization using commercial softwares.

关键词： Clustering Finite element method Numerical homogenization

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OPTIMAL CONTROL FOR MULTISCALE ELLIPTIC EQUATIONS WITH ROUGH COEFFICIENTS

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Journal of Computational Mathematics 2023年第5期41卷 841-865页

作者： Yanping Chen Xinliang Liu Jiaoyan Zeng Lei Zhang School of Mathematical Sciences South China Normal UniversityGuangzhou 510631China King Abdullah University of Science and Technology Thuwal 23955Saudi Arabia School of Mathematics and Statistics Guangdong University of FinanceGuangzhou 510631China School of Mathematical Sciences Institute of Natural Sciencesand MOE-LSCShanghai Jiao Tong UniversityShanghai 200240China

This paper concerns the convex optimal control problem governed by multiscale elliptic equations with arbitrarily rough L∞coefficients,which has not only complex coupling between nonseparable scales and nonlinearity,but also important applications in composite materials and geophysics.We use one of the recently developed Numerical homogenization techniques,the so-called Rough Polyharmonic Splines(RPS)and its generalization(GRPS)for the efficient resolution of the elliptic operator on the coarse scale.Those methods have optimal convergence rate which do not rely on the regularity of the coefficients nor the concepts of scale-separation or periodicity.As the iterative solution of the nonlinearly coupled OCP-OPT formulation for the optimal control problem requires solving the corresponding(state and co-state)multiscale elliptic equations many times with different right hand sides,Numerical homogenization approach only requires one-time pre-computation on the fine scale and the following iterations can be done with computational cost proportional to coarse degrees of freedom.numerical experiments are presented to validate the theoretical analysis.

关键词： Optimal control Rough coefficients Multiscale elliptic equations Numerical homogenization Rough polyharmonic splines Iterative algorithm

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COMPUTATIONAL MULTISCALE METHODS FOR LINEAR HETEROGENEOUS POROELASTICITY

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Journal of Computational Mathematics 2020年第1期38卷 41-57页

作者： Robert Altmann Eric Chung Roland Maier Daniel Peterseim Sai-Mang Pun Department of Mathematics University of AugsburgAugsburgGermany Department of Mathematics The Chinese University of Hong KongHong KongChina Department of Mathematics Texas UniversityUSA

We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling.This poroelasticity problem suffers from rapidly oscillating material parameters,which calls for a thorough numerical treatment.In this paper,we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity.Therein,local corrector problems are constructed in line with the static equations,whereas we propose to consider the full system.This allows to benefit from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure.We prove the optimal first-order convergence of this method and verify the result by numerical experiments.

关键词： Poroelasticity Heterogeneous media Numerical homogenization Multiscale methods

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