热弹性动力学耦合问题的插值型移动最小二乘无网格法研究
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摘要
该文基于插值型移动最小二乘法,将无网格局部Petrov-Galerkin(MLPG)法用于二维耦合热弹性动力学问题的求解。修正的Fourier热传导方程和弹性动力控制方程通过加权余量法来离散,Heaviside分段函数作为局部弱形式的权函数,从而得到描述热耦合问题的二阶常微分方程组。然后利用微分代数方法,温度和位移作为辅助变量,将上述二阶常微分方程组转换成常微分代数系统,采用Newmark逐步积分法进行求解。该方法无需Laplace变换可直接得到温度场和位移场数值结果,同时插值型移动最小二乘法构造的形函数由于满足Kronecker delta特性,因此能直接施加本质边界条件。最后通过两个数值算例来验证该方法的有效性。
The two-dimensional structural dynamic coupled thermoelastic problem is solved by meshless local Petrov-Galerkin(MLPG) method based on the interpolating moving least-squares(IMLS) method. The local weak forms are developed using the weighted residual method from the modified Fourier heat conduction equations and elastodynamic equations, in which the Heaviside step function is used as the test function in each sub-domain.Then the second-order ordinary differential equations describing the coupled thermoelasticity problem are obtained. Using the differential algebraic method, these second-order ordinary differential equations can be transformed into ordinary differential algebraic systems, in which temperature and displacement are chosen as auxiliary variables. The Newmark step-integration method is used to solve the ordinary differential system. The temperature and displacement numerical results can be obtained directly without the Laplace transform. Since the shape functions constructed from the IMLS method possess the Kronecker delta property, the essential boundary conditions can be implemented directly. Finally, two numerical examples are studied to illustrate the effectiveness of this method.
The two-dimensional structural dynamic coupled thermoelastic problem is solved by meshless local Petrov-Galerkin(MLPG) method based on the interpolating moving least-squares(IMLS) method. The local weak forms are developed using the weighted residual method from the modified Fourier heat conduction equations and elastodynamic equations, in which the Heaviside step function is used as the test function in each sub-domain.Then the second-order ordinary differential equations describing the coupled thermoelasticity problem are obtained. Using the differential algebraic method, these second-order ordinary differential equations can be transformed into ordinary differential algebraic systems, in which temperature and displacement are chosen as auxiliary variables. The Newmark step-integration method is used to solve the ordinary differential system. The temperature and displacement numerical results can be obtained directly without the Laplace transform. Since the shape functions constructed from the IMLS method possess the Kronecker delta property, the essential boundary conditions can be implemented directly. Finally, two numerical examples are studied to illustrate the effectiveness of this method.
引文
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[16]Sladek J,Sladek V,Zhang C,et al.Meshless local Petrov-Galerkin method for linear coupled thermoelastic analysis[J].Computer modeling in Engineering and Sciences,2006,16(1):57―68.
[17]Hosseini S M,Sladek J,Sladek V.Meshless local Petrov-Galerkin method for coupled thermoelasticity analysis of a functionally graded thick hollow cylinder[J].Engineering Analysis with Boundary Elements,2011,35(6):827―835.
[18]Zheng B J,Dai B D.A meshless local moving Kriging method for two-dimensional solids[J].Applied Mathematics and Computation,2011,218(2):563―573.
[19]王峰.改进的无网格法求解温度场和温度应力及其在水工结构分析中的应用[D].大连:大连理工大学,2015.Wang Feng.Improved meshless method for the solution of temperature field and thermal stress and its application to hydraulic structure analysis[D].Dalian:Dalian University of Technology,2015.(in Chinese)
[20]Zheng B J,Gao X W,Yang K,et al.A novel meshless local Petrov-Galerkin method for dynamic coupled thermoelasticity analysis under thermal and mechanical shock loading[J].Engineering Analysis with Boundary Elements,2015,60:154―161.
[21]程玉民.移动最小二乘法研究进展与述评[J].计算机辅助工程,2009,18(2):5―11.Cheng Yumin.Advances and review on moving leastsquare methods[J].Computer Aided Engineering,2009,18(2):5―11.(in Chinese)
[22]Lancaster P,Salkauskas K.Surface generated by moving least squares methods[J].Mathematics of computation,1981,37(155):141―158.
[23]任红萍,程玉民,张武.弹性力学的插值型边界无单元法[J].中国科学:物理学力学天文学,2010,40(3):361―369.Ren Hongping,Cheng Yumin,Zhang Wu.Interpolating boundary element free method for elasticity[J].Chinese Science:Physics,mechanics and Astronomy,2010,40(3):361―369.(in Chinese)
[2]Mavri?B,?arler B.Application of the RBF collocation method to transient coupled thermoelasticity[J].International Journal of Numerical Methods for Heat&Fluid Flow,2017,27(5):1064―1077.
[3]Nowinski J L.Theory of thermoelasticity with applications[M].New York:Sijhoff&Noordhoff International Publishers,1978.
[4]Boley B A,Weiner J H.Theory of thermal stresses[M].New York:Dover Publications,2011.
[5]马永斌,何天虎.基于分数阶热弹性理论的含有球型空腔无限大体的热冲击动态响应[J].工程力学,2016,33(7):31―38.Ma Yongbin,He Tianhu.Thermal shock dynamic response of an infinite body with a spherical cavity under fractional order theory of thermoelasticity[J].Engineering Mechanics,2016,33(7):31―38.(in Chinese)
[6]谷良贤,王一凡.几何非线性假设下温度大范围变化瞬态热力耦合问题研究[J].工程力学,2016,33(8):221―230.Gu Liangxian,Wang Yifan.The transient response of thermo-mechanical coupling with wide change in temperature based on the hypothesis of geometry nonliearity[J].Engineering Mechanics,2016,33(8):221―230.(in Chinese)
[7]Boley B A,Tolins I S.Transient coupled thermoelastic boundary value problems in the half-space[J].Journal of Applied Mechanics-ASME,1962,29(4):637―646.
[8]Nariboli G A.Spherically symmetric thermal shock in a medium with thermal and elastic deformations coupled[J].The Quarterly Journal of Mechanics and Applied Mathematics,1959,14(1):75―84.
[9]Takeuti Y,Ishida R,Tanigawa Y.On an axisymmetric coupled thermal stress problem in a finite circular cylinder[J].Journal of Applied Mechanics-ASME,1983,50(1):116―122.
[10]Oden J T.Finite elements of nonlinear continua[M].New York:Dover Publications,2006.
[11]Park K H,Banerjee P K.Two-and three-dimensional transient thermoelastic analysis by BEM via particular integrals[J].International Journal of Solids and Structures,2002,39(10):2871―2892.
[12]Tehrani P H,Eslami M R.BEM analysis of thermal and mechanical shock in a two-dimensional finite domain considering coupled thermoelasticity[J].Engineering Analysis with Boundary Elements,2000,24(3):249―257.
[13]王林翔,梅尔姆克.热弹性动力学耦合问题的微分代数方法[J].应用数学和力学,2006,27(9):1036―1046.Wang Linxiang,Melnik R V N.Differential algebraic approach to coupled problems of dynamic thermoelasticity[J].Applied Mathematics and Mechanics,2006,27(9):1036―1046.(in Chinese)
[14]Atluri S N,Zhu T.A new meshless local Petrov-Galerkin(MLPG)approach in computational mechanics[J].Computational Mechanics,1998,22(2):117―127.
[15]Atluri S N,Shen S P.The meshless local petrov-galerkin(MLPG):a simple&less-costly alternative to the finite element and boundary element methods[J].Computer modeling in Engineering and Sciences,2002,3(1):11―51.
[16]Sladek J,Sladek V,Zhang C,et al.Meshless local Petrov-Galerkin method for linear coupled thermoelastic analysis[J].Computer modeling in Engineering and Sciences,2006,16(1):57―68.
[17]Hosseini S M,Sladek J,Sladek V.Meshless local Petrov-Galerkin method for coupled thermoelasticity analysis of a functionally graded thick hollow cylinder[J].Engineering Analysis with Boundary Elements,2011,35(6):827―835.
[18]Zheng B J,Dai B D.A meshless local moving Kriging method for two-dimensional solids[J].Applied Mathematics and Computation,2011,218(2):563―573.
[19]王峰.改进的无网格法求解温度场和温度应力及其在水工结构分析中的应用[D].大连:大连理工大学,2015.Wang Feng.Improved meshless method for the solution of temperature field and thermal stress and its application to hydraulic structure analysis[D].Dalian:Dalian University of Technology,2015.(in Chinese)
[20]Zheng B J,Gao X W,Yang K,et al.A novel meshless local Petrov-Galerkin method for dynamic coupled thermoelasticity analysis under thermal and mechanical shock loading[J].Engineering Analysis with Boundary Elements,2015,60:154―161.
[21]程玉民.移动最小二乘法研究进展与述评[J].计算机辅助工程,2009,18(2):5―11.Cheng Yumin.Advances and review on moving leastsquare methods[J].Computer Aided Engineering,2009,18(2):5―11.(in Chinese)
[22]Lancaster P,Salkauskas K.Surface generated by moving least squares methods[J].Mathematics of computation,1981,37(155):141―158.
[23]任红萍,程玉民,张武.弹性力学的插值型边界无单元法[J].中国科学:物理学力学天文学,2010,40(3):361―369.Ren Hongping,Cheng Yumin,Zhang Wu.Interpolating boundary element free method for elasticity[J].Chinese Science:Physics,mechanics and Astronomy,2010,40(3):361―369.(in Chinese)