Efficient Finite Element Time Domain Method Based On Compressive Sensing

Analysis of multiscale and complex structure over a broad frequency band is currently a hot issue and a hard problem in electromagnetic field theory and its engineering application.Since it is flexible in geometric modeling by using unstructured grid,the finite element time domain(FETD)method has natural advantages in handling this problem.But a large system of linear equations is still required to be solved at each time step in the FETD method,which always leads to a high computational complexity and limits its application in analysis of multiscale and complex problems.With the unremitting efforts of researchers,the performance of the FETD method has been improved,and the discontinuous Galerkin time domain(DGTD)method which is more efficient than the FETD is developed on this basis.In the DGTD method,the fields in two adjacent elements are connected through the numerical flux on each face.As a result,the fields can be updated just by solving the independent small matrix equations in every element.However,due to the solution of equations at each time step,the calculation amount of long-time simulation is also huge in both FETD and DGTD method.In order to reduce the computational complexity of solution of matrix equations,a class of accelerated solution methods is proposed by introducing the compressive sensing(CS)theory to the FETD method in this dissertation,and the effectiveness and advantages of the proposed methods are validated by the analysis of computational complexity and numerical examples.The main work and innovation are as following:(1)An accelerated iteration model of implicit solution is provided for FETD and a new sparse transform scheme is constructed.The determined system of equations is transformed to underdetermined system of equations by extracting the rows from the mass matrix in the FETD method based on CS theory,then the efficient FETD implicit solution model is built.Additionally,to efficiently solve the implicit solution model by applying recovery algorithm,a new sparse transform is devised in which the results of previous time steps are taken as prior knowledge according to the characteristics of the time domain method.(2)An efficient explicit solution model for the FETD method is proposed,and a restart mechanism to maintain the accuracy is formed.With help of the recovery algorithm in the CS frame,the electromagnetic field is represented as a linear combination of the results of previous time steps.Then the efficient explicit solution model is derived by substituting the representation into the time stepping matrix equations of the FETD method.Directly solving matrix equations is avoided in the model,and the updating of field can be calculated just by one multiplication of matrix and vector.Furthermore,the restart mechanism is built for the explicit solution model to control the computational complexity and the accumulated error.(3)A global solution method of the DGTD is proposed based on CS theory.In this method,firstly,the matrix equations in every element is combined into a global matrix equation which is in a block diagonal form,and then a global solution model is built by applying the scheme similar to the implicit solution model;Secondly,to reduce the computational complexity,a sparse transform scheme which is constituted by progressive electromagnetic field of previous time steps and a subspace transform scheme which is constituted by stepping electromagnetic field of previous time steps are put forward;Finally,the global analysis results of electromagnetic field are efficiently obtained by recovery algorithms and the least square method with the progressive scheme and the stepping scheme respectively.As a result,the limits on computational complexity of implicit solution and explicit solution method which is caused by the conventional FETD method can be broken though in the global solution method of DGTD.