Translated Abstract
People are used to study natural phenomenon by classifying them into different subjects. Based on this, the cognizing system of single physical field is built. However, most of the physical processes in nature exist in the form of two-way coupling multi-physics. Therefore, in order to grasp the physical and mathematical fundamentals of these physical process, people have to apply effective analysis and calculation method.
Analysis methods of multi-physics problems can be generally categorized into the direct coupling method and the indirect coupling method. While using the latter to deal with the two-way coupling multi-physics, we solve every single physical field one by one, then update one physical field every time, and do interation between them until all solutions are convergent. This method has poor convergence and sometimes can’t converge. However, by sovling all degrees of freedom(DOFs) simultaneously, the direct coupling method could improve the convergence of solving the two-way coupling multi-physics. Therefore, this thesis focuses on the direct coupling method of the electro-magneto-thermal multi-physics. The main work of this thesis can be summarizes as follows.
The quantities in different subjects such as electricity, magnetics, thermodynamics and so on, are classified by using an analogue method. An unified physical description is built based on the difference of the intensive quantities and the flow of the extensive quantities. Then the analogy of constitutive relation and energy reppresentation in different subjects is analyzed. Based on the analogy, an unified modeling approach based on the cell method is studied.
By transforming the difference of the intensive quantities and the flow of the extensive quantities into their intergral forms, the modeling approach eliminates the impact on the topological equations caused by the mesh size. Thus, a calculation method with consistency is built for all different fields of physics.
It is analyzed and proved that the discretization error of the constitutive equations is the major error source of the cell method, which is closedly related to the relative position between the difference of the intensive quantities and the flow of the extensive quantities in the same cell. Based on a posteriori error estimation, a constrained optimization problem is built with the goal of reducing the discretization error of the constitutive equations. Then an algorithm of moving the dual mesh is proposed by using the support vector machines and the patch recovery technology. The numerical test demonstrates that this self-adaptive cell method could achieve the goal that high-precision numerical solutions can be obtained under the corase mesh, which establishes a theoretical and technical foundation for solving the multi-physical spatial multi-scale model.
The electro-magneto-thermal non-linear coupling model is build and analyzed. Then a direct coupling calculation method base on the Jacobian-free Newton-GMRES(JFNG) is proposed.
According to the feature that the coupling model has muti degrees of freedom, a coupling operator splitting preconditioning method is proposed. This preconditioning method could accelerate the convergence rate of JFNG by improving the spectrum distribution of the coefficient matrix. Since the coupling operator splitting preconditioning method turns the product operation between the subblock of the Jacobian matrix and a given vector into a difference operation, so the Jacobian matrix of the nonlinear system doesn’t need to be calculated and preconditioned explicitly, which saves a large amount of memory.
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