Domain Decomposition for Electromagnetic Field Simulation
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Solution Overview
Problem
Current domain decomposition methods for simulating electromagnetic fields face challenges in achieving high convergence rates across a broad range of applications, particularly due to the limitations of first-order Robin transmission conditions which only damp propagation errors and not evanescent errors, and higher-order conditions suffer from poor convergence properties due to sensitivity to Robin coefficients.
Innovation Solution
The use of higher-order Robin transmission conditions, optimized for fast convergence by selecting suitable Robin coefficients based on mesh size, frequency of electromagnetic fields, and basis order of finite elements, along with a domain-decomposition formulation using block lower triangular subdomain matrices and a modified Gauss-Seidel preconditioner for iterative solution.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If first-order Robin transmission conditions are used, then the mathematical problem is well-posed and convergence occurs, but the convergence rate is slow because propagation errors are damped but evanescent errors are not
Solution Approach 1:
The patent applies parameter changes by transitioning from first-order to higher-order Robin transmission conditions. Specifically, second-order Robin conditions are used to damp both propagation and evanescent errors, thereby improving the convergence rate while maintaining the well-posedness of the mathematical problem. This parameter change in the order of the transmission condition directly addresses the contradiction between reliability and productivity.
2Productivity
If higher-order Robin transmission conditions are used, then the convergence rate may be improved, but the convergence becomes highly sensitive to the choice of Robin coefficients
Solution Approach 1:
The patent systematically changes the parameters (Robin coefficients) based on the specific problem characteristics such as frequency, mesh size, and material properties. By establishing systematic methods to select these parameters, the patent reduces the sensitivity to coefficient choice while maintaining the high convergence rate benefits of higher-order conditions.
Solution Approach 2:
The patent incorporates feedback mechanisms where the Robin coefficients are selected based on problem-specific parameters and convergence behavior. This feedback approach allows the method to adapt to different scenarios, reducing sensitivity to arbitrary coefficient choices while maintaining optimal convergence rates.
3Measurement precision
If direct solvers are used for large three-dimensional problems, then an exact solution can be obtained, but prohibitive amounts of memory are required
Solution Approach 1:
The patent applies segmentation by dividing the large three-dimensional domain into multiple subdomains. This domain decomposition transforms the single large linear system into multiple smaller subsystems that can be solved more efficiently. The segmentation enables the use of iterative solvers with reduced memory requirements while maintaining solution accuracy through proper interface conditions.
Solution Approach 2:
The patent substitutes the direct solver approach (mechanical system) with an iterative solver approach. This substitution replaces the memory-intensive direct inversion with an iterative process that uses less memory, particularly when combined with domain decomposition and appropriate preconditioning strategies.
4Quantity of substance
If iterative solvers are used for large systems, then memory requirements are reduced, but the number of iterations required for convergence increases
Solution Approach 1:
The patent uses domain decomposition to segment the problem into subdomains, which enables the use of efficient iterative solvers on each subdomain. This segmentation, combined with appropriate preconditioning, reduces the number of iterations required for convergence while maintaining low memory requirements.
Solution Approach 2:
The patent optimizes convergence by changing parameters such as using higher-order transmission conditions and selecting appropriate Robin coefficients based on problem characteristics. These parameter changes accelerate convergence, reducing the number of iterations and thus the computational time while maintaining the memory efficiency of iterative solvers.
Data Source
AI summary
Disclosed are domain-decomposition approaches to simulations of electromagnetic fields may that, in various embodiments, use second-order Robin transmission conditions at subdomain boundaries.