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Why is the boundary element method so powerful in fluid mechanics? Uncover its mathematical foundation!
In recent years, the boundary element method (BEM) has been hotly discussed in fluid mechanics and other fields. As a numerical calculation method, BEM is changing the way we analyze fluid behavior with its simplified calculation requirements and effective boundary processing technology. This method not only improves computational efficiency, but also makes it possible to handle complex boundary conditions. The mathematical basis behind it is worth exploring.
The boundary element method is a numerical calculation method for solving linear partial differential equations. It converts the problem into a boundary integral equation, which is especially suitable in fluid mechanics.
The core idea of the boundary element method is to focus on boundary conditions rather than the values of the entire space. In this way, BEM simplifies the problems that need to be dealt with to only the boundaries. Such a transformation means a significant reduction in the amount of data, which has greater advantages especially in problems with higher dimensions. When the boundary conditions are accurately embedded in the integral equation, the equation can be used in the post-processing stage to numerically calculate the solution anywhere inside.
It is worth noting that BEM is suitable for problems where green functions are computable. This is common in many linear homogeneous media, but also limits the scope of application of these methods. For nonlinear problems, although it can be incorporated into the setting of the method, it will introduce volume integration, which requires the discretization of the volume, which affects the initial superiority of BEM. In response to this, the dual-reciprocity method was proposed to handle volume integrals in a way that does not require discretizing the volume. This method converts the volume integral into a boundary integral through a local interpolation function.
In double reciprocal BEM, the unknowns within the selected points are included in the linear algebra equation, making the solution of the problem more convenient.
The boundary element method also faces numerical computational challenges, especially when the distance between the source point and the target element is large. At this point, conventional green function integration becomes difficult, especially when the system equations are based on singular loads (e.g., electric fields from point charges). Although analytical integration is possible for simple element geometries such as planar triangles, general elements often require purely numerical schemes designed for singularities, which significantly increases computational cost. In response to these problems, improving the speed and efficiency of boundary element problem calculation has become a current research hotspot.
The advantage of BEM is that it exhibits higher computational efficiency than other methods in certain specific cases. For example, in problems with small surface/volume ratios, the boundary element method has demonstrated its high efficiency, but in many cases, compared with volume discretization methods (such as finite element methods or finite difference methods), advanced BEM may not be able to achieve Same efficiency.
For example, when a liquid tumbles in a storage tank, the boundary element method can efficiently calculate its natural frequency and achieve accurate numerical simulations.
In addition, the boundary element method usually produces a full matrix, which means that as the size of the problem grows, its storage requirements and calculation time increase quadratically. In contrast, finite element matrices are usually band-shaped, which makes their storage requirements grow linearly with problem size. While certain compression techniques can alleviate this problem, their application is complex and their effectiveness varies depending on problem characteristics and geometry.
Taken together, the boundary element method is undoubtedly a powerful tool for solving fluid mechanics problems. It provides a more concise and efficient solution in many cases, especially in specific problems. However, such technology still requires continuous exploration and innovation when faced with nonlinear problems and computational efficiency challenges.
In the context of today's rapid development of numerical simulation technology, how will the boundary element method compete with other numerical methods and continue to evolve?
