The Research And Application Of The Invariance Of The Orthogonal Complement Space In The Projection Decomposition Method

Computational electromagnetics is a crosscutting field based on mathematical methods and computer technology for the study of electromagnetic field theory and numerical solutions.It has been widely used in military and civilian fields such as communications,radar,electromagnetic compatibility,medical diagnosis,etc.,which has made it possible to efficiently and quickly solve large-scale and complex electromagnetic field problems.In particular,for the optimization of large-scale complex multi-scale electromagnetic structures,in addition to the changes in the optimized part,the remaining parts often do not change,and the invariance of this physical geometry is not reflected in the electromagnetic simulation algorithm,instead,the fact is to calculate these invariant structures repeatedly.Through the study of this paper,the projective decomposition method is used to solve the small-scale problem of dividing a large-scale problem into several sub-regions.It can not only make the large-scale problem solvable,but also separate the optimization region from the invariant region to solve separately.It is also possible to separate the large-scale target and the small-scale target from the multi-scale structure,and use the coarse grid and the fine grid separately.In this way,the calculation scale of the problem can be reduced,the calculation efficiency can be improved,and the accuracy of the numerical solution can be improved too.For the invariant region in the optimization problem,this paper proposes a projection decomposition method based on orthogonal complement space invariance.The algorithm preserves the orthogonal complement space corresponding to the invariant region obtained in the projection decomposition method based on the orthogonal complement space.In solving partial structural optimization problems,the relative invariance of its orthogonal complement space can be used to construct a new orthogonal complement space directly,and the solution to the new problem can be obtained in one iteration.The correctness of the algorithm is strictly proved in mathematics,and it is explained by algebraic and geometric methods.Then through a number of numerical examples verify the high efficiency of the algorithm,usually the method can save more than 99% of the calculation time.In addition,for large-scale scattering problems,in order to set the absorption boundary close to the target body to reduce the size of the unknown,a method of fitting the MEI coefficients using coefficients of the second-order precision discrete format of the first-order Mur absorption boundary is proposed,including the fitting formulas.Then the validity of the fitting coefficient is verified by an example,which greatly reduces the calculation scale and calculation amount.Finally,in order to combine the compatible sub-gridding method and the projection decomposition method based on orthogonal complementary space more effectively,the discrete format of the first type of transition nodes in the sub-gridding method is improved,and a new discrete format is proposed.A numerical example is used to verify the computational efficiency and numerical accuracy of the new format,and the number of iterations of the projection decomposition method is significantly reduced.Then,the compatibility sub-gridding technology and the orthogonal decomposition-space-relative invariance projection decomposition algorithm are combined to apply to SIW waveguide and filter optimization design calculations,verifying the effectiveness,high accuracy and high efficiency of the new algorithm in solving large-scale electromagnetic field problems.