比例边界有限元法在势流理论中的应用
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摘要
在计算流体力学研究领域,许多数值模拟方法已经非常成熟,如有限差分法,有限元法,边界元法,有限体积法等。近些年采用无网格计算的粒子法也得到了广泛应用。比例边界有限元法是一种新的数值方法,最早应用在固体力学领域,它是一种结合了有限元法和边界元法优点的数值模拟方法。在应用比例边界有限元法进行计算时,采用了一个新的坐标系,即辐射坐标和环向坐标。单元离散的时同边界元法一样只需要在计算区域的边界环向坐标方向上进行离散,大大减少了计算量。此外比例边界有限元法在其辐射坐标上保持了解析性,因此其模拟精度较高。该方法的另一个优点是可以自动满足无限远的边界条件。比例边界有限元法是一种只在计算区域的边界上离散单元的有限元法,在数值模拟领域有着广泛的应用空间。
本文应用比例边界有限元法对势流理论的问题进行了数值模拟,包括有限域和无限域的势流运动问题、二维和三维的波浪与结构物作用问题、二维和三维容器内液体晃荡问题,均获得了比较好的数值模拟结果。
本文首先以二维势流问题为例,以Laplace方程为基本控制方程,分别叙述了有限域情况和无限域情况下比例边界有限元方程的建立和求解过程,并讨论了在计算时比例中心位置选取的问题。对于波浪与二维结构物的相互作用问题,以波浪与水面单个固定方箱相互作用为例,在采用比例边界有限元法模拟的时候,划分了两个有限子域和两个无限子域,详细推导了两种不同类型子域中比例边界有限元方程的建立和求解过程,并将数值模拟结果与解析解和边界元法数值模拟结果进行了对比,证明了本文采用的比例边界有限元法的精确性和高效性。随后讨论了计算网格的选取和计算区域选取对计算精度的影响。通过改变有限域中比例中心的位置,比例边界有限元法可以用于模拟多种形式的结构物,如潜堤、薄板防波堤等。对不同尺寸的结构物进行了模拟,并将模拟结构进行对比,给出了其一般的变化规律。通过与解析解和其他数值方法模拟结果以及试验结果的对比,证明了本文方法的精确性和高效性。
随后通过改变比例中心位置,改进了应用比例边界有限元在模拟二维容器内液体晃荡问题的算法。通常的做法是将比例中心设置在自由水面上,这样导致了求解过程变得很繁琐,将比例中心设置在容器底部或者计算域内时,求解过程大大简化,同时也提高了模拟的精度。对矩形容器内的液体晃荡问题,还计算出了其内部的速度势分布和速度分布,与解析解都十分吻合。而后推导了三维容器内液体晃荡问题的比例边界有限元方程及其求解过程,分别用线性单元和高阶单元进行数值模拟,通过与解析解的对比证明了高阶单元模拟的精度要更高。又对于圆柱形容器内的液体晃荡问题进行模拟,数值模拟结果与解析解吻合也较好。
最后将比例边界有限元法与特征函数展开法耦合来模拟波浪与结构物的作用。对于二维问题,在有限子域中采用比例边界有限元法,在无限子域中采用特征函数展开的方法,并在其交界上匹配来求解,通过对比证明采用这样的方法可以提高计算的效率和精度。对于波浪与三维结构物的作用问题,本文以波浪与垂直圆柱相互作用为例进行模拟,并与解析解进行了对比,证明了将比例边界有限元法与特征函数展开法联合求解的可行性与精确性。
In the field of computational fluid dynamics,there are many numerical simulation methods,such as the finite difference method,the finite element method,the boundary element method,the finite volume method,etc.Recently,some new methods have been used in this field,such as particle method.The scaled boundary element finite element method is a novel method,which combines the advantages of the finite element and the boundary element methods,and this method was used in the structural mechanics firstly.A new coordinate system including the circumferential local co-ordinate and the radial co-ordinate has been established in the scaled boundary element finite element method.Only the boundary of the computational domain needs to be discretized in the circumferential direction as the same as the boundary element method.The solution in the radial direction is analytical,so the simulation precision of this method is high.This method can meet the infinity of the boundary condition automatically.The scaled boundary element finite element method is the finite element method which is discredited on the boundary of the computational domain,and it has a wide range of applications in the field of numerical simulation.
In this paper the potential flow problems have been simulated using the scaled boundary element finite element method,including the potential flow in bounded domain and unbounded domain,the 2D or 3D wave action on structures and the water sloshing in 2D or 3D containers.
Firstly the Laplace equation has been solved with the scaled boundary element finite element method for the potential flow problems.The derivative and solution process have been given for both the bounded domain and the unbounded domain problems,and the location of the scaled center has been discussed.For the problem wave action on 2D structures,the wave action on a fixed box on the free surface has been simulated using this method.The derivative and solution process have been given for both bound domain problems and unbound domain problems,and the simulation results have been compared with the analytic solution and the results from a boundary element method.The accuracy and the efficiency have been confirmed.Then the effect of the selection of grid and the distance of the computational domain has been discussed.This method can be used in solving the wave interaction with various forms of structure,such as submerged breakwaters,barrier breakwaters and so on,by changing the position of scaled center in bound domain.The wave action with structures for different sizes has been simulated,and the changing characters of the simulation results have been given.Through comparing with the analytic solution,the results from other numerical methods and experimental results,the accuracy and efficiency of this method have been testified.
The water sloshing in 2D containers is studied by the scaled boundary finite element method.The usual method is to set the scaled center on the water surface for this problem, which results the calculation complicately.By changing the proportion of the scaled center, the solution process has been simplified,and the accuracy of simulation has been improved. Then the water sloshing in 2D Semi-circular and rectangular containers is studied,and the distribution of velocity potential and velocity has been given,which agree well with the analytical solutions.At last the 3D water sloshing problem is studied.The derivative and solution process of the scaled boundary finite element method have been derived in detail. The liner element and the high order element have been used in the simulation.Through comparison with the 2D analytical solution,it is found that the simulation with the high order element has a higher precision.The cylindrical container has also been computed,and the simulation results are agree with the analytical solution well.
Finally,the scaled boundary finite element method has been coupled with the eigenfunction expansion method to simulate the wave reaction with structure problem.For the 2D problem,the scaled boundary finite element method is used in the bounded domain and the eigenfunction expansion method is used in the unbounded domain,and coupling on the junction.This method has higher precision and efficiency.For the 3D problem,the wave reaction with cylinder has been simulated,and the accuracy and efficiency of this method have been testified through the comparison with the analytic solutions.
本文应用比例边界有限元法对势流理论的问题进行了数值模拟,包括有限域和无限域的势流运动问题、二维和三维的波浪与结构物作用问题、二维和三维容器内液体晃荡问题,均获得了比较好的数值模拟结果。
本文首先以二维势流问题为例,以Laplace方程为基本控制方程,分别叙述了有限域情况和无限域情况下比例边界有限元方程的建立和求解过程,并讨论了在计算时比例中心位置选取的问题。对于波浪与二维结构物的相互作用问题,以波浪与水面单个固定方箱相互作用为例,在采用比例边界有限元法模拟的时候,划分了两个有限子域和两个无限子域,详细推导了两种不同类型子域中比例边界有限元方程的建立和求解过程,并将数值模拟结果与解析解和边界元法数值模拟结果进行了对比,证明了本文采用的比例边界有限元法的精确性和高效性。随后讨论了计算网格的选取和计算区域选取对计算精度的影响。通过改变有限域中比例中心的位置,比例边界有限元法可以用于模拟多种形式的结构物,如潜堤、薄板防波堤等。对不同尺寸的结构物进行了模拟,并将模拟结构进行对比,给出了其一般的变化规律。通过与解析解和其他数值方法模拟结果以及试验结果的对比,证明了本文方法的精确性和高效性。
随后通过改变比例中心位置,改进了应用比例边界有限元在模拟二维容器内液体晃荡问题的算法。通常的做法是将比例中心设置在自由水面上,这样导致了求解过程变得很繁琐,将比例中心设置在容器底部或者计算域内时,求解过程大大简化,同时也提高了模拟的精度。对矩形容器内的液体晃荡问题,还计算出了其内部的速度势分布和速度分布,与解析解都十分吻合。而后推导了三维容器内液体晃荡问题的比例边界有限元方程及其求解过程,分别用线性单元和高阶单元进行数值模拟,通过与解析解的对比证明了高阶单元模拟的精度要更高。又对于圆柱形容器内的液体晃荡问题进行模拟,数值模拟结果与解析解吻合也较好。
最后将比例边界有限元法与特征函数展开法耦合来模拟波浪与结构物的作用。对于二维问题,在有限子域中采用比例边界有限元法,在无限子域中采用特征函数展开的方法,并在其交界上匹配来求解,通过对比证明采用这样的方法可以提高计算的效率和精度。对于波浪与三维结构物的作用问题,本文以波浪与垂直圆柱相互作用为例进行模拟,并与解析解进行了对比,证明了将比例边界有限元法与特征函数展开法联合求解的可行性与精确性。
In the field of computational fluid dynamics,there are many numerical simulation methods,such as the finite difference method,the finite element method,the boundary element method,the finite volume method,etc.Recently,some new methods have been used in this field,such as particle method.The scaled boundary element finite element method is a novel method,which combines the advantages of the finite element and the boundary element methods,and this method was used in the structural mechanics firstly.A new coordinate system including the circumferential local co-ordinate and the radial co-ordinate has been established in the scaled boundary element finite element method.Only the boundary of the computational domain needs to be discretized in the circumferential direction as the same as the boundary element method.The solution in the radial direction is analytical,so the simulation precision of this method is high.This method can meet the infinity of the boundary condition automatically.The scaled boundary element finite element method is the finite element method which is discredited on the boundary of the computational domain,and it has a wide range of applications in the field of numerical simulation.
In this paper the potential flow problems have been simulated using the scaled boundary element finite element method,including the potential flow in bounded domain and unbounded domain,the 2D or 3D wave action on structures and the water sloshing in 2D or 3D containers.
Firstly the Laplace equation has been solved with the scaled boundary element finite element method for the potential flow problems.The derivative and solution process have been given for both the bounded domain and the unbounded domain problems,and the location of the scaled center has been discussed.For the problem wave action on 2D structures,the wave action on a fixed box on the free surface has been simulated using this method.The derivative and solution process have been given for both bound domain problems and unbound domain problems,and the simulation results have been compared with the analytic solution and the results from a boundary element method.The accuracy and the efficiency have been confirmed.Then the effect of the selection of grid and the distance of the computational domain has been discussed.This method can be used in solving the wave interaction with various forms of structure,such as submerged breakwaters,barrier breakwaters and so on,by changing the position of scaled center in bound domain.The wave action with structures for different sizes has been simulated,and the changing characters of the simulation results have been given.Through comparing with the analytic solution,the results from other numerical methods and experimental results,the accuracy and efficiency of this method have been testified.
The water sloshing in 2D containers is studied by the scaled boundary finite element method.The usual method is to set the scaled center on the water surface for this problem, which results the calculation complicately.By changing the proportion of the scaled center, the solution process has been simplified,and the accuracy of simulation has been improved. Then the water sloshing in 2D Semi-circular and rectangular containers is studied,and the distribution of velocity potential and velocity has been given,which agree well with the analytical solutions.At last the 3D water sloshing problem is studied.The derivative and solution process of the scaled boundary finite element method have been derived in detail. The liner element and the high order element have been used in the simulation.Through comparison with the 2D analytical solution,it is found that the simulation with the high order element has a higher precision.The cylindrical container has also been computed,and the simulation results are agree with the analytical solution well.
Finally,the scaled boundary finite element method has been coupled with the eigenfunction expansion method to simulate the wave reaction with structure problem.For the 2D problem,the scaled boundary finite element method is used in the bounded domain and the eigenfunction expansion method is used in the unbounded domain,and coupling on the junction.This method has higher precision and efficiency.For the 3D problem,the wave reaction with cylinder has been simulated,and the accuracy and efficiency of this method have been testified through the comparison with the analytic solutions.
引文
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