电磁场无网格分析方法及纸基高频感应封合系统研究
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摘要
无网格法无需采用网格近似,对于计算时模型几何尺寸经常发生变化的问题,可以避免网格基方法所存在的网格畸变或重复划分网格的困扰。很多电磁问题,如含刚体运动的电磁场问题、电磁场逆问题、金属融化和铸造问题,薄板以及小空气间隙等问题,采用网格基方法时也面临着网格畸变或网格重复剖分的瓶颈,故无网格法作为一种潜在能解决上述问题的方法,引起了众多学者的注意。当前,电磁场计算的难点和热点问题在于与工程应用相结合的运动电磁问题和耦合电磁场问题的研究。纸基复合材料感应封合系统是一个含运动导体的电、磁、热相耦合的多材料非线性系统,具备可靠封合效果是无菌灌装设备封合系统的关键要求。本文针对无网格法在电磁场计算研究中的几个基本问题如数值模型建立方法、交界面条件处理和含运动的电磁场无网格法等问题,进行了系统的研究,并在对无菌饮料灌装的纸基复合材料感应封合问题进行研究和模拟的基础上,设计出了具有可靠封合效果的感应封合系统。
电磁场控制方程无网格法数值模型的建立是应用无网格法开展电磁问题研究的基础。文章对Maxwell方程组做一定假设,引入磁矢量位(A)和电标量位(φ),建立由A-φ-A描述的谐波电磁场控制方程。然后,针对一大类可以采用圆柱坐标系表示的电磁问题,采用矢量检验函数和标量检验函数,应用Galerkin加权余量法和无网格迦辽金法形函数对控制方程进行离散,详细推导出了谐波电磁场在圆柱坐标系下的无网格迦辽金法频域数值模型,为后续应用算例的研究打下了良好基础。
无网格法对于电磁场边、交界条件,尤其是交界面条件处理的不足是限制无网格法在电磁场计算中应用的关键问题。文章在对无网格法中处理不连续性问题的方法进行分析的基础上,提出采用分割加修改弱形式法如Lagrange乘子法和罚函数法处理电磁场交界面条件,并就处理过程进行描述和推导,建立其处理模型。所采用的几个应用算例证实了所建立的谐波电磁场无网格法数值模型的正确性,表明无网格法可以有效用于电磁场的数值计算;所建议的电磁场交界面条件处理方法可以引入必要的不连续性,也可以弥补产生期待的连续性。指出罚函数法施加电磁场交界面条件的精度取决于罚系数的取值,并给出了罚系数的合理取值;而Lagrange乘子法施加交界面条件的精度取决于交界面上插值点的数目,数目越多,越精确,但过多插值点将导致系统矩阵的奇异。
含刚体运动的电磁问题是计算电磁学的重要问题。文章提出采用无网格法求解带速度项的电磁场控制方程,发现速度和控制节点影响域尺寸大小的参数dmax是使运动电磁场无网格法解中包含数值伪振荡的主要因素。加密节点可以抑制带速度项电磁场控制方程的无网格法解中的数值伪振荡,但随速度和dmax的增大,所需节点数也越多,计算量也越大。因此,文章将streamline迎风稳定方法引入到运动电磁场无网格法中,指出无网格法中迎风稳定因子不再是优化的,并详细推导出了带速度项电磁场控制方程的streamline迎风无网格迦辽金法数值模型。所采用的两个应用算例证实streamline迎风法和无网格迦辽金法的结合可以有效求解带速度项的电磁场控制方程,证实了所建立的数值模型的正确性。尽管迎风稳定因子不再是优化的,但streamline迎风仍然能显著抑制运动导体电磁场无网格法解中的数值伪振荡,得到稳定的解。
纸基复合材料(包材)高频感应封合过程是电、磁、热相耦合的过程。文章在分析影响封合效果因素的基础上,对感应封合系统进行设计,建立封合系统的电磁场无网格法模型,并就电磁场与温度场的耦合进行有限元模拟,研究不同因素对包装材料中涡流分布、温度分布以及所需加热时间的影响,得到感应封合无网格法解的参考解。所设计的封合系统能在封合线上得到均匀的温度分布。激励电流大小是影响封合的主要因素,大电流短时间加热比小电流长时间加热更有效;激励电流频率的增大有助于涡流密度和热量向封合线集中。磁通集中器是实现包材有效封合必不可少的因素,它可显著提高包材封合线及厚度方向上的涡流和温度,从而提高封合速度和效率。在进行上述模拟和分析的基础上,给出了适合纸基复合材料感应横封的封合参数。砖包感应横封实验表明所建议的封合参数的正确性。
In computational electromagnetics, for such problems that the model geometry changes frequently in computation process as electromagnetic field involving moving conductor, the electromagnetic inverse problem, the thin plate problem, and the melting and casting of metal, the grids-based method faces the bottleneck of re-meshing and grid-distortion, so the meshless method due to no grids needed is looked as a potential method to solve the above electromagnetic problems, and attracts the attentions of some researchers. At present, the difficult point and hotspot in computational electromagnetics lie in the research on the coupling or moving electromagnetic problem combined with engineering design. The induction sealing system of paper-based composite material is a nonlinear multi-material system coupling electric, magnetic and thermo. The reliable sealing effect is the key requirement of the induction sealing system in aseptic beverage packaging machine. This paper aims at a few basic problems of meshless method in computational electromagnetic field, including the method for building meshless numerical model, the disposal of interface condition, the meshless method in moving electromagnetic field and so on, systematic researches were carried out. At the same time, based on the study and simulation on the induction sealing problem of paper-based composite material in aseptic beverage packaging machine, an induction sealing system with reliable sealing effect was designed.
The foundation of meshless numerical model of electromagnetic field governing equation is the basis of the application of meshless method in electromagnetic problem. Omitting the displacement current and accumulation charge in Maxwell equations and introducing the magnetic vector potential A and the electric scalar potentialφ, the harmonic electromagnetic field governing equation formulated by A-φ-A approach was completed. Then, aiming at the electromagnetic problem which can be expressed in cylindrical coordinates, using the vector and scalar test function, the detailed implementation of the governing equation was described with the meshless method, and the common 3D numerical model in frequency domain was derived by using Galerkin residual method.
The disposal of boundary and interface conditions, especially the interface conditions, is the key problem to restrict the application of the meshless method in numerical computation of engineering electromagnetic field. Based on the analysis about the present methods for the discontinuity disposal in meshless method, this paper presents such methods based on the modification of the weak form as Lagrange multiplier method and penalty method to deal with electromagnetic interface conditions. Using above two methods, the detailed disposal process and models of the electromagnetic interface conditions were described further in the meshless method. The application examples used prove that the above meshless method numerical model of electromagnetic field is corrected, and the meshless method is found to be reliable to give reasonable computational results for electromagnetic field; the suggested methods for enforcing the electromagnetic interfaces can introduce the necessary discontinuity and also guarantee the expected continuity; the disposal accuracy of penalty method for electromagnetic interfaces is decided by the value of penalty coefficient, and the suitable value is (1e2-1e5)×(Kii)max; the disposal accuracy of Lagrange multiplier method for electromagnetic interfaces is decided by the number of the interpolation point at interface, the number is larger, the results are more accurate, but too large point number would cause deteriorated result, and even a singular system matrix
The numerical modeling of rigid body motion is an important problem in computational electromagnetics. In this paper, the meshless method was proposed to solve the electromagnetic field governing equation with velocity term, and the computational result indicates that the velocity and the parameter dmax controlling the size of nodal influence domain are the major factors to lead to the numerical oscillation contained in the meshless method solution. The nodal refinement method used can suppress the numerical oscillation, but with the increasing velocity and dmax, the nodal number increases. So, the streamline upwinding stabilization method in FEM was introduced into the meshless method for electromagnetic field involving moving conductor, and the streamline upwinding meshless method numerical model was derived for electromagnetic governing equation with velocity term formulated by A-φ-A method. Different numerical examples show that the combination streamline upwinding and the meshless method can solve effectively the electromagnetic governing equation with velocity term. Although the stabilization factor is no longer optimal in meshless method, the streamline upwinding can still suppress significantly the numerical oscillation in meshless method solution of electromagnetic field governing equation with velocity term.
The induction sealing system of paper-based composite material is a nonlinear system coupling electric, magnetic and thermo. In this paper, based on analysis on the factors influencing sealing effect, an induction sealing system was designed, the meshless method model of sealing electromagnetic field was founded, and the coupling of electromagnetic and thermal field was simulated by FEM. The influences of different factors to heating time, eddy current and temperature distribution were studied, and the reference solution of the meshless method solution of sealing system was obtained. The uniform temperature in sealing line can be obtained by the designed induction sealing system. The exciting current is the major factor influencing the seal, the instant seal only can be realized by large current; the larger the current frequency is, the more the eddy current density and Joule heat concentrate to the seal line. The magnetic flux concentrator is necessary to realize the effective and fast sealing of packaging material, and it can increase significantly the eddy current and temperature at the seal line and thickness direction of packaging material. Based on above analyses and simulations, the suitable sealing parameters of paper-based composite material are given, the experiments prove that the proposed sealing parameters are correct.
电磁场控制方程无网格法数值模型的建立是应用无网格法开展电磁问题研究的基础。文章对Maxwell方程组做一定假设,引入磁矢量位(A)和电标量位(φ),建立由A-φ-A描述的谐波电磁场控制方程。然后,针对一大类可以采用圆柱坐标系表示的电磁问题,采用矢量检验函数和标量检验函数,应用Galerkin加权余量法和无网格迦辽金法形函数对控制方程进行离散,详细推导出了谐波电磁场在圆柱坐标系下的无网格迦辽金法频域数值模型,为后续应用算例的研究打下了良好基础。
无网格法对于电磁场边、交界条件,尤其是交界面条件处理的不足是限制无网格法在电磁场计算中应用的关键问题。文章在对无网格法中处理不连续性问题的方法进行分析的基础上,提出采用分割加修改弱形式法如Lagrange乘子法和罚函数法处理电磁场交界面条件,并就处理过程进行描述和推导,建立其处理模型。所采用的几个应用算例证实了所建立的谐波电磁场无网格法数值模型的正确性,表明无网格法可以有效用于电磁场的数值计算;所建议的电磁场交界面条件处理方法可以引入必要的不连续性,也可以弥补产生期待的连续性。指出罚函数法施加电磁场交界面条件的精度取决于罚系数的取值,并给出了罚系数的合理取值;而Lagrange乘子法施加交界面条件的精度取决于交界面上插值点的数目,数目越多,越精确,但过多插值点将导致系统矩阵的奇异。
含刚体运动的电磁问题是计算电磁学的重要问题。文章提出采用无网格法求解带速度项的电磁场控制方程,发现速度和控制节点影响域尺寸大小的参数dmax是使运动电磁场无网格法解中包含数值伪振荡的主要因素。加密节点可以抑制带速度项电磁场控制方程的无网格法解中的数值伪振荡,但随速度和dmax的增大,所需节点数也越多,计算量也越大。因此,文章将streamline迎风稳定方法引入到运动电磁场无网格法中,指出无网格法中迎风稳定因子不再是优化的,并详细推导出了带速度项电磁场控制方程的streamline迎风无网格迦辽金法数值模型。所采用的两个应用算例证实streamline迎风法和无网格迦辽金法的结合可以有效求解带速度项的电磁场控制方程,证实了所建立的数值模型的正确性。尽管迎风稳定因子不再是优化的,但streamline迎风仍然能显著抑制运动导体电磁场无网格法解中的数值伪振荡,得到稳定的解。
纸基复合材料(包材)高频感应封合过程是电、磁、热相耦合的过程。文章在分析影响封合效果因素的基础上,对感应封合系统进行设计,建立封合系统的电磁场无网格法模型,并就电磁场与温度场的耦合进行有限元模拟,研究不同因素对包装材料中涡流分布、温度分布以及所需加热时间的影响,得到感应封合无网格法解的参考解。所设计的封合系统能在封合线上得到均匀的温度分布。激励电流大小是影响封合的主要因素,大电流短时间加热比小电流长时间加热更有效;激励电流频率的增大有助于涡流密度和热量向封合线集中。磁通集中器是实现包材有效封合必不可少的因素,它可显著提高包材封合线及厚度方向上的涡流和温度,从而提高封合速度和效率。在进行上述模拟和分析的基础上,给出了适合纸基复合材料感应横封的封合参数。砖包感应横封实验表明所建议的封合参数的正确性。
In computational electromagnetics, for such problems that the model geometry changes frequently in computation process as electromagnetic field involving moving conductor, the electromagnetic inverse problem, the thin plate problem, and the melting and casting of metal, the grids-based method faces the bottleneck of re-meshing and grid-distortion, so the meshless method due to no grids needed is looked as a potential method to solve the above electromagnetic problems, and attracts the attentions of some researchers. At present, the difficult point and hotspot in computational electromagnetics lie in the research on the coupling or moving electromagnetic problem combined with engineering design. The induction sealing system of paper-based composite material is a nonlinear multi-material system coupling electric, magnetic and thermo. The reliable sealing effect is the key requirement of the induction sealing system in aseptic beverage packaging machine. This paper aims at a few basic problems of meshless method in computational electromagnetic field, including the method for building meshless numerical model, the disposal of interface condition, the meshless method in moving electromagnetic field and so on, systematic researches were carried out. At the same time, based on the study and simulation on the induction sealing problem of paper-based composite material in aseptic beverage packaging machine, an induction sealing system with reliable sealing effect was designed.
The foundation of meshless numerical model of electromagnetic field governing equation is the basis of the application of meshless method in electromagnetic problem. Omitting the displacement current and accumulation charge in Maxwell equations and introducing the magnetic vector potential A and the electric scalar potentialφ, the harmonic electromagnetic field governing equation formulated by A-φ-A approach was completed. Then, aiming at the electromagnetic problem which can be expressed in cylindrical coordinates, using the vector and scalar test function, the detailed implementation of the governing equation was described with the meshless method, and the common 3D numerical model in frequency domain was derived by using Galerkin residual method.
The disposal of boundary and interface conditions, especially the interface conditions, is the key problem to restrict the application of the meshless method in numerical computation of engineering electromagnetic field. Based on the analysis about the present methods for the discontinuity disposal in meshless method, this paper presents such methods based on the modification of the weak form as Lagrange multiplier method and penalty method to deal with electromagnetic interface conditions. Using above two methods, the detailed disposal process and models of the electromagnetic interface conditions were described further in the meshless method. The application examples used prove that the above meshless method numerical model of electromagnetic field is corrected, and the meshless method is found to be reliable to give reasonable computational results for electromagnetic field; the suggested methods for enforcing the electromagnetic interfaces can introduce the necessary discontinuity and also guarantee the expected continuity; the disposal accuracy of penalty method for electromagnetic interfaces is decided by the value of penalty coefficient, and the suitable value is (1e2-1e5)×(Kii)max; the disposal accuracy of Lagrange multiplier method for electromagnetic interfaces is decided by the number of the interpolation point at interface, the number is larger, the results are more accurate, but too large point number would cause deteriorated result, and even a singular system matrix
The numerical modeling of rigid body motion is an important problem in computational electromagnetics. In this paper, the meshless method was proposed to solve the electromagnetic field governing equation with velocity term, and the computational result indicates that the velocity and the parameter dmax controlling the size of nodal influence domain are the major factors to lead to the numerical oscillation contained in the meshless method solution. The nodal refinement method used can suppress the numerical oscillation, but with the increasing velocity and dmax, the nodal number increases. So, the streamline upwinding stabilization method in FEM was introduced into the meshless method for electromagnetic field involving moving conductor, and the streamline upwinding meshless method numerical model was derived for electromagnetic governing equation with velocity term formulated by A-φ-A method. Different numerical examples show that the combination streamline upwinding and the meshless method can solve effectively the electromagnetic governing equation with velocity term. Although the stabilization factor is no longer optimal in meshless method, the streamline upwinding can still suppress significantly the numerical oscillation in meshless method solution of electromagnetic field governing equation with velocity term.
The induction sealing system of paper-based composite material is a nonlinear system coupling electric, magnetic and thermo. In this paper, based on analysis on the factors influencing sealing effect, an induction sealing system was designed, the meshless method model of sealing electromagnetic field was founded, and the coupling of electromagnetic and thermal field was simulated by FEM. The influences of different factors to heating time, eddy current and temperature distribution were studied, and the reference solution of the meshless method solution of sealing system was obtained. The uniform temperature in sealing line can be obtained by the designed induction sealing system. The exciting current is the major factor influencing the seal, the instant seal only can be realized by large current; the larger the current frequency is, the more the eddy current density and Joule heat concentrate to the seal line. The magnetic flux concentrator is necessary to realize the effective and fast sealing of packaging material, and it can increase significantly the eddy current and temperature at the seal line and thickness direction of packaging material. Based on above analyses and simulations, the suitable sealing parameters of paper-based composite material are given, the experiments prove that the proposed sealing parameters are correct.
引文
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