复变量重构核粒子法研究
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摘要
无网格方法是近年来发展起来的一种新兴的数值方法,因其不需要网格,只需要节点信息,具有前处理简单、计算精度高等特点,已成为目前科学和工程计算方法的研究热点之一,也是科学和工程计算发展的趋势。
重构核粒子法是目前应用和研究比较广泛的无网格方法之一。本文针对目前重构核粒子法配点过多、计算量大等问题,提出了复变量重构核粒子法,然后将其应用于势问题、瞬态热传导问题、弹性力学、弹性动力学和弹塑性力学等,并研究了复变量重构核粒子法和有限元的耦合法。具体研究工作如下:
在重构核粒子法的基础上,本文提出了复变量重构核粒子法,推导了复变量重构核粒子法公式。与传统的重构核粒子法相比,复变量重构核粒子法的优点是在形函数的构造中采用一维基函数建立二维问题的修正函数,使得修正函数中所含的待定系数减少,从而有效提高计算效率。
将复变量重构核粒子法应用于势问题,提出了势问题的复变量重构核粒子法,推导了相应的计算公式。该方法的优点是可取较少的节点,在同等精度下,相比传统的重构核粒子法减小了计算量;而在同等节点分布时,相比传统的重构核粒子法提高了精度。
在稳态热传导问题的基础上,将复变量重构核粒子法应用于瞬态热传导问题的求解,结合瞬态热传导问题的Galerkin积分弱形式,建立了瞬态热传导问题的复变量重构核粒子法,推导了相应的计算公式。
将复变量重构核粒子法应用于弹性力学问题,对其控制方程的等效积分弱形式,采用罚函数法施加本质边界条件,建立了弹性力学的复变量重构核粒子法,推导了相应的计算公式。弹性力学的复变量重构核粒子法具有求解精度高、可消除体积闭锁现象等优点。
将复变量重构核粒子法推广应用于求解弹性动力学问题,由Galerkin积分弱形式得到离散系统求解方程,采用Newmark时间积分方案,并采用罚函数法施加本质边界条件,建立了弹性动力学的复变量重构核粒子法,推导了相应的公式。
在弹性力学的复变量重构核粒子法的基础上,在小变形假设的前提下,采用增量形式的复变量重构核粒子法进行插值,利用增量形式的应力应变关系表征材料的弹塑性本构关系,采用罚因子修正能量变分方程式以施加本质边界条件,数值实现中采用了Newton-Raphson增量迭代法,提出了基于增量本构关系的弹塑性力学的复变量重构核粒子法。算例表明,复变量重构核粒子法在求解弹塑性问题时具有稳定性好、收敛快的优点。
由于复变量重构核粒子法的形函数不具有Kronecker Delta函数特性,因此边界条件的处理是复变量重构核粒子法实施中的一个难点。本文将复变量重构核粒子法和有限元法进行了耦合,提出了势问题和弹性力学的复变量重构核粒子法与有限元的耦合法。用该耦合法在进行势问题和弹性力学问题分析时不仅可以方便地施加本质边界条件,而且可以充分利用复变量重构核粒子法和有限元法的优势,弥补各自不足以提高计算效率。
为了证明本文提出的复变量重构核粒子法的有效性,本文编制了MATLAB计算程序,进行了数值算例分析。数值算例说明了本文方法的正确性和有效性。
In recent years,a class of new numerical methods called meshless methods has been proposed.The meshless methods only need the information at nodes,and don't discretize the domain into a mesh.The advantages of meshless methods are that the pre-processing is simple,and they have high precision.These features make the meshless methods a hot point and the development trend of numerical methods for science and engineering problems.
The reproducing particle method(RKPM) is one,which is studied and applied widely,of the meshless methods.But it has great computation cost because of large numbers of nodes selected in the domain of problem.Then in this dissertation,on the basis of the RKPM,the complex variable reproducing kernel particle method (CVRKPM) is developed,and is applied to solve potential problems,transient heat conduction problems,elasticity problems,elastodynamics problems and elastoplasticity problems respectively.At last,the coupled FEM and CVRKPM for analyzing potential and elasticity problems are presented.The main researches of this thesis are as follows.
On the basis of the RKPM,the CVRKPM is presented in this paper.The formations of the CVRKPM are obtained in detail.Comparing to the RKPM,the advantage of the CVRKPM is that the correction function of a 2D problem is formed with 1D basis function when the shape function is formed.Then the unknown coefficients of correction function in the CVRKPM are fewer than in the RKPM.And then the computational efficiency of the CVRKPM is greater.
The CVRKPM is applied to two-dimensional potential problems,and the CVRKPM for potential problem is presented,and the corresponding formulae are obtained.Compared with the RKPM,under the same precision,the CVRKPM can select fewer nodes,and then has greater computational efficiency.On the other hand, under the same nodes,the CVRKPM has greater precision than the RKPM.
On the basis of the steady heat conduction problems,the CVRKPM is applied to two-dimensional transient heat conduction problems.Combining the Galerkin weak form of transient heat conduction problems,the CVRKPM for transient heat conduction problems is investigated.And the corresponding formulae are obtained.
The CVRKPM is applied to two-dimensional elasticity.The discrete equation is produced from the weak form of variational equation,the penalty parameters are used to enforce the essential boundary conditions,and then the CVRKPM for elasticity is presented.And the corresponding formulae are obtained.The advantages of this method are higher precision,and that volume closure phenomena can be avoided.
The CVRKPM is applied to two-dimensional elastodynamics.The Galerkin weak form of elastodynamics problems is employed to obtain the discretized system equations,and the Newmark time integration method is used for time history analyses. And the penalty method is employed to apply the essential boundary conditions.Then the CVRKPM for elastodynamics is presented.And the corresponding formulae are obtained.
On the basis of the CVRKPM for elasticity,when the small deformation is assumed,the incremental complex variable reproducing particle approximation is adopted,the increments of stress and strain are used to characterize the elastoplastic constitutive relationship,the penalty parameters are adopted to revise the energy variation equations to enforce the essential boundary conditions,and the Newton-Raphson iteration techniques are introduced into the numerical implementation, then CVRKPM for elastoplasticity problems is proposed.Numerical examples show that the CVRKPM for elastoplasticity problems has the advantages of good stability and higher convergence speed.
As the shape function of the CVRKPM does not satisfy the property of Kronecker Delta function,which makes it difficult to impose the essential boundary condition of the problem.By combining the CVRKPM and finite element method(FEM),a coupled method of FEM and CVRKPM for analyzing two-dimensional elasticity or potential problems is presented.The coupled method not only emerge the essential boundary condition conveniently,but also exploit their advantages while avoiding their disadvantages to have higher computation efficiency.
In order to show the efficiency of the CVRKPM in the dissertation,the MATLAB codes of the methods above have been written.Some numerical examples are provided, and the validity and efficiency of these methods are demonstrated.
重构核粒子法是目前应用和研究比较广泛的无网格方法之一。本文针对目前重构核粒子法配点过多、计算量大等问题,提出了复变量重构核粒子法,然后将其应用于势问题、瞬态热传导问题、弹性力学、弹性动力学和弹塑性力学等,并研究了复变量重构核粒子法和有限元的耦合法。具体研究工作如下:
在重构核粒子法的基础上,本文提出了复变量重构核粒子法,推导了复变量重构核粒子法公式。与传统的重构核粒子法相比,复变量重构核粒子法的优点是在形函数的构造中采用一维基函数建立二维问题的修正函数,使得修正函数中所含的待定系数减少,从而有效提高计算效率。
将复变量重构核粒子法应用于势问题,提出了势问题的复变量重构核粒子法,推导了相应的计算公式。该方法的优点是可取较少的节点,在同等精度下,相比传统的重构核粒子法减小了计算量;而在同等节点分布时,相比传统的重构核粒子法提高了精度。
在稳态热传导问题的基础上,将复变量重构核粒子法应用于瞬态热传导问题的求解,结合瞬态热传导问题的Galerkin积分弱形式,建立了瞬态热传导问题的复变量重构核粒子法,推导了相应的计算公式。
将复变量重构核粒子法应用于弹性力学问题,对其控制方程的等效积分弱形式,采用罚函数法施加本质边界条件,建立了弹性力学的复变量重构核粒子法,推导了相应的计算公式。弹性力学的复变量重构核粒子法具有求解精度高、可消除体积闭锁现象等优点。
将复变量重构核粒子法推广应用于求解弹性动力学问题,由Galerkin积分弱形式得到离散系统求解方程,采用Newmark时间积分方案,并采用罚函数法施加本质边界条件,建立了弹性动力学的复变量重构核粒子法,推导了相应的公式。
在弹性力学的复变量重构核粒子法的基础上,在小变形假设的前提下,采用增量形式的复变量重构核粒子法进行插值,利用增量形式的应力应变关系表征材料的弹塑性本构关系,采用罚因子修正能量变分方程式以施加本质边界条件,数值实现中采用了Newton-Raphson增量迭代法,提出了基于增量本构关系的弹塑性力学的复变量重构核粒子法。算例表明,复变量重构核粒子法在求解弹塑性问题时具有稳定性好、收敛快的优点。
由于复变量重构核粒子法的形函数不具有Kronecker Delta函数特性,因此边界条件的处理是复变量重构核粒子法实施中的一个难点。本文将复变量重构核粒子法和有限元法进行了耦合,提出了势问题和弹性力学的复变量重构核粒子法与有限元的耦合法。用该耦合法在进行势问题和弹性力学问题分析时不仅可以方便地施加本质边界条件,而且可以充分利用复变量重构核粒子法和有限元法的优势,弥补各自不足以提高计算效率。
为了证明本文提出的复变量重构核粒子法的有效性,本文编制了MATLAB计算程序,进行了数值算例分析。数值算例说明了本文方法的正确性和有效性。
In recent years,a class of new numerical methods called meshless methods has been proposed.The meshless methods only need the information at nodes,and don't discretize the domain into a mesh.The advantages of meshless methods are that the pre-processing is simple,and they have high precision.These features make the meshless methods a hot point and the development trend of numerical methods for science and engineering problems.
The reproducing particle method(RKPM) is one,which is studied and applied widely,of the meshless methods.But it has great computation cost because of large numbers of nodes selected in the domain of problem.Then in this dissertation,on the basis of the RKPM,the complex variable reproducing kernel particle method (CVRKPM) is developed,and is applied to solve potential problems,transient heat conduction problems,elasticity problems,elastodynamics problems and elastoplasticity problems respectively.At last,the coupled FEM and CVRKPM for analyzing potential and elasticity problems are presented.The main researches of this thesis are as follows.
On the basis of the RKPM,the CVRKPM is presented in this paper.The formations of the CVRKPM are obtained in detail.Comparing to the RKPM,the advantage of the CVRKPM is that the correction function of a 2D problem is formed with 1D basis function when the shape function is formed.Then the unknown coefficients of correction function in the CVRKPM are fewer than in the RKPM.And then the computational efficiency of the CVRKPM is greater.
The CVRKPM is applied to two-dimensional potential problems,and the CVRKPM for potential problem is presented,and the corresponding formulae are obtained.Compared with the RKPM,under the same precision,the CVRKPM can select fewer nodes,and then has greater computational efficiency.On the other hand, under the same nodes,the CVRKPM has greater precision than the RKPM.
On the basis of the steady heat conduction problems,the CVRKPM is applied to two-dimensional transient heat conduction problems.Combining the Galerkin weak form of transient heat conduction problems,the CVRKPM for transient heat conduction problems is investigated.And the corresponding formulae are obtained.
The CVRKPM is applied to two-dimensional elasticity.The discrete equation is produced from the weak form of variational equation,the penalty parameters are used to enforce the essential boundary conditions,and then the CVRKPM for elasticity is presented.And the corresponding formulae are obtained.The advantages of this method are higher precision,and that volume closure phenomena can be avoided.
The CVRKPM is applied to two-dimensional elastodynamics.The Galerkin weak form of elastodynamics problems is employed to obtain the discretized system equations,and the Newmark time integration method is used for time history analyses. And the penalty method is employed to apply the essential boundary conditions.Then the CVRKPM for elastodynamics is presented.And the corresponding formulae are obtained.
On the basis of the CVRKPM for elasticity,when the small deformation is assumed,the incremental complex variable reproducing particle approximation is adopted,the increments of stress and strain are used to characterize the elastoplastic constitutive relationship,the penalty parameters are adopted to revise the energy variation equations to enforce the essential boundary conditions,and the Newton-Raphson iteration techniques are introduced into the numerical implementation, then CVRKPM for elastoplasticity problems is proposed.Numerical examples show that the CVRKPM for elastoplasticity problems has the advantages of good stability and higher convergence speed.
As the shape function of the CVRKPM does not satisfy the property of Kronecker Delta function,which makes it difficult to impose the essential boundary condition of the problem.By combining the CVRKPM and finite element method(FEM),a coupled method of FEM and CVRKPM for analyzing two-dimensional elasticity or potential problems is presented.The coupled method not only emerge the essential boundary condition conveniently,but also exploit their advantages while avoiding their disadvantages to have higher computation efficiency.
In order to show the efficiency of the CVRKPM in the dissertation,the MATLAB codes of the methods above have been written.Some numerical examples are provided, and the validity and efficiency of these methods are demonstrated.
引文
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