Data-driven Discovery of Multiple-Physics Electromagnetic Partial Differential Equations

The subject of data-driven discovery for equations has developed rapidly in recent years, especially in the field of finding equations of unknown forms, which provides new ideas for the study of complex systems. When there are unknown noise sources and other uncertain factors in the system, it is quite difficult to directly derive the system governing equation, because the equation is complicated and the calculation cost is large. But if we try to find the equation directly from the data, it will be helpful to improve these problems. For the data in nonlinear multi-physics electromagnetic system, the deep learning method can be used to find the equation, which can obtain the governing equation form accurately and has high time efficiency and parameter precision. This thesis studies the algorithm of data-driven discovery equations in electromagnetic multiple physics problems and realizes the inversion of Maxwell's multiple physics equations. Firstly, three methods of data-driven equation discovery are introduced, including symbol regression, sparse regression and neural network. Secondly, an algorithm based on sparse regression and convolutional neural network is proposed for multiple physics equations of Maxwell equations. This algorithm uses Euler method to approximate time differentiation and convolution kernel to compute space differentiation. At the same time, in the training process, the pareto analysis method was used to remove the redundancy. Then, the model algorithm is applied to the multi-physics coupling simulation data of electromagnetic plasma, and the homogeneous and non-homogeneous equations of electromagnetic propagation are realized by using less time and space observation field samples, which has certain anti-noise performance. For the problem of propagation in uniform medium, the influence of spatial and temporal sampling method on the inversion precision of equation coefficients is studied. Under the condition of inhomogeneous media propagation, this thesis finds the changing law of inhomogeneous coefficient by changing the weight scale of neural network, aiming at the problem that the equation coefficient varies with the spatial scale. By using the properties of trigonometric series and some prior knowledge, the expression of the coefficient of inhomogeneous terms is approximated, and satisfactory results are obtained. Finally, the thesis summarizes the proposed method and its main conclusion. In both homogeneous and inhomogeneous media, the model has good performance. Meanwhile, the author discusses the possible improvement methods for other problems and the idea that the structure of the model can be adjusted in a small range in the future and applied to the high-dimensional space and the problems with high-order spatial differentiation in the governing equations

On the eigenvalues of a specially rank-r updated complex matrix

AbstractIn this paper, an alternatively simpler proof to an eigenvalue theorem of a specially structured rank-r updated complex matrix is presented and also its characteristic polynomial is explicitly determined by Leverrier’s algorithm for m–D system

Optimal aerodynamic design of hypersonic inlets by using streamline-tracing techniques

Rectangular-to-Ellipse Shape Transition (REST) inlets are a class of inward turning inlets designed for hypersonic flight. The aerodynamic design of REST inlets involves very complex flows and shock-wave patterns. These inlets are used in highly integrated propulsive systems. Often the design of these inlets may require many geometrical constraints at different cross-section. In present work a design approach for hypersonic inward-turning inlets, adapted for REST inlets, is coupled with a multi-objective optimization procedure. The automated procedure iterates on the parametric representation and on the numerical solution of a base flow from which the REST inlet is generated by using streamline tracing and shape transition algorithms. The typical design problem of optimizing the total pressure recovery and mass flow capture of the inlet is solved by the proposed procedure. The accuracy of the optimal solutions found is discussed and the performances of the designed REST inlets are investigated by means of fully 3-D Euler and 3-D RANS analyses

Aerodynamic Design of Inward-Turning Inlets with Shape Transition

Rectangular-to-Ellipse Shape Transition (REST) inlets are a kind of inward-turning inlets designed for hypersonic vehicles, especially under integration design backgrounds. The streamline tracing technique is an inverse method for designing inward-turning inlets by extracting different streamtubes from the same reference flow. In present work, the streamline tracing technique is coupled with an optimization procedure. The procedure for designing a REST inlet with prescribed mass capture at the design point and optimal performance is illustrated