功能梯度材料的多尺度建模
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摘要
本文主要研究梯度功能材料（FGM）的多尺度建模。梯度功能材料是一种可以应用于多个领域的复合材料。我们首先在均匀化理论框架下，对于具有周期性微观结构的复合材料，由微观模型利用渐进展开的方法得到宏观模型，并给出了该多尺度模型相应的收敛性分析。然后，对于具有一般微观结构的FGM，在异质多尺度方法（HMM）框架下，给出了由微观模型估计宏观参数的算法。在材料具有微观周期结构的情况下，我们给出了其HMM解和均匀化解之间的误差分析。最后，我们给出一个算例，即对某种给定的复合材料分别用上述两种方法得到宏观模型，并求解比较以验证我们的结论。
In this paper, we considered the multiscale modelling and simulation offunctionally graded materials(FGM). FGM is a composite materials which could beapplied in many areas. First of all, we derived the macroscopic model from themicroscopic one for the materials with periodic microstructure. We carried out thecorresponding convergence analysis under the frame of homogenization. For gen-gral cases, we desigened the multiscale model under the frame of HeterogeneousMultiscale Method(HMM) and gave out the error estimate between the HMM so-lutions and the homogenization solutions under the assumption that there exists aperiodic microscale structure. At last, numerical experiments were carried out todemonstrate the accuracy of the methods.
In this paper, we considered the multiscale modelling and simulation offunctionally graded materials(FGM). FGM is a composite materials which could beapplied in many areas. First of all, we derived the macroscopic model from themicroscopic one for the materials with periodic microstructure. We carried out thecorresponding convergence analysis under the frame of homogenization. For gen-gral cases, we desigened the multiscale model under the frame of HeterogeneousMultiscale Method(HMM) and gave out the error estimate between the HMM so-lutions and the homogenization solutions under the assumption that there exists aperiodic microscale structure. At last, numerical experiments were carried out todemonstrate the accuracy of the methods.
引文
[1]边洁,梯度功能材料的研究进展,山东机械,2002(3):2-4.
[2] Y.P.Feng, J.Z.Cui, Multiscale analysis for the structure of composite materialswith small periodic confguration under condition of thermo―elasticity, ActaMechanical Sinica，2003(6)：585―592．
[3] Y.P.Feng, J.Z.Cui, Multiscale analysis and FE Computation for the structureof composite materials with small periodic confguration under condition ofcoupled Thermoelasticity, Intem J Numer Meth Engg,2004(8):1879―1910．
[4]张晓超,拟周期结构复合材料的二阶双尺度渐近分析,广州:广州大学硕士论文,2010.
[5]万建军,张同斌,周期复合材料等效热弹性性能的数值计算,河南科技大学学报,2009年8月,第30卷第4期:12-15.
[6]万建军,肖留超,刘鸣放,复合材料耦合热弹性问题的多尺度方法,暨南大学学报,2010年6月,第31卷第3期:235-237.
[7]万建军,复合材料结构瞬态热力耦合问题多尺度分析方法,北京:中科院博士学位论文,2007．
[8]肖纯,热弹性方程的均匀化,湛江师范学院学报,2011年12月,第32卷第6期:55-60.
[9] Horia I.Ene, On linear thermoelasticity of composite materials, In-t.J.Engng.Sci.,1983,21.5:443-448.
[10]张同斌,万建军,非均匀材料耦合热弹性问题解的存在唯一性,河南大学学报,2009年5月,第39卷第3期:231-234.
[11]孙涛,宋士仓,小周期复合材料波传播问题的一个多尺度渐近展开式及其收敛性分析,工程数学学报,2009,26(1)
[12]王自强,宋士仓,曹俊英,小周期复合材料热传导问题的双尺度渐近展开及收敛性分析,高校应用数学学报A辑,2008,23(2)
[13] W.E and B.Engquist, The heterogeneous multi-scale methods, Com-m.Math.Sci.,2003,1:87-133.
[14] W.E and P.B. Ming and P.W. Zhang, Analysis of the heterogeneous multi-scale method for elliptic homogenization problems, J.Amer.Math.Soc.,2003,18(1):121-156.
[15] P.G.Clarlet, The fnite element method for elliptic problems, SIAM,2002
[16] A.Abedulle and M.J.Grote, Finite element heterogeneous multiscale method forthe wave equation, SIAM Multi.Model.Simul,2010
[17] S.I Chou and C.C.Wang, Estimates of error in fnite element approximate solu-tions to problems in linear thermoelasticity, Springer Berlin/Heidelberg,1981.
[18] V.V.Jikov, S.M.Kozlov, O.A.Oleinik, Homogenization of Diferential Operatorsand Integral Functions, Springer-Verlag,1994.
[2] Y.P.Feng, J.Z.Cui, Multiscale analysis for the structure of composite materialswith small periodic confguration under condition of thermo―elasticity, ActaMechanical Sinica，2003(6)：585―592．
[3] Y.P.Feng, J.Z.Cui, Multiscale analysis and FE Computation for the structureof composite materials with small periodic confguration under condition ofcoupled Thermoelasticity, Intem J Numer Meth Engg,2004(8):1879―1910．
[4]张晓超,拟周期结构复合材料的二阶双尺度渐近分析,广州:广州大学硕士论文,2010.
[5]万建军,张同斌,周期复合材料等效热弹性性能的数值计算,河南科技大学学报,2009年8月,第30卷第4期:12-15.
[6]万建军,肖留超,刘鸣放,复合材料耦合热弹性问题的多尺度方法,暨南大学学报,2010年6月,第31卷第3期:235-237.
[7]万建军,复合材料结构瞬态热力耦合问题多尺度分析方法,北京:中科院博士学位论文,2007．
[8]肖纯,热弹性方程的均匀化,湛江师范学院学报,2011年12月,第32卷第6期:55-60.
[9] Horia I.Ene, On linear thermoelasticity of composite materials, In-t.J.Engng.Sci.,1983,21.5:443-448.
[10]张同斌,万建军,非均匀材料耦合热弹性问题解的存在唯一性,河南大学学报,2009年5月,第39卷第3期:231-234.
[11]孙涛,宋士仓,小周期复合材料波传播问题的一个多尺度渐近展开式及其收敛性分析,工程数学学报,2009,26(1)
[12]王自强,宋士仓,曹俊英,小周期复合材料热传导问题的双尺度渐近展开及收敛性分析,高校应用数学学报A辑,2008,23(2)
[13] W.E and B.Engquist, The heterogeneous multi-scale methods, Com-m.Math.Sci.,2003,1:87-133.
[14] W.E and P.B. Ming and P.W. Zhang, Analysis of the heterogeneous multi-scale method for elliptic homogenization problems, J.Amer.Math.Soc.,2003,18(1):121-156.
[15] P.G.Clarlet, The fnite element method for elliptic problems, SIAM,2002
[16] A.Abedulle and M.J.Grote, Finite element heterogeneous multiscale method forthe wave equation, SIAM Multi.Model.Simul,2010
[17] S.I Chou and C.C.Wang, Estimates of error in fnite element approximate solu-tions to problems in linear thermoelasticity, Springer Berlin/Heidelberg,1981.
[18] V.V.Jikov, S.M.Kozlov, O.A.Oleinik, Homogenization of Diferential Operatorsand Integral Functions, Springer-Verlag,1994.