What are the applications and limitations of MLPG for fluid dynamics?

1 What is MLPG?

MLPG is a meshfree method that does not rely on a predefined mesh or grid to discretize the fluid domain. Instead, it uses a set of scattered nodes (or particles) that can be randomly distributed or arranged according to some criteria. MLPG applies the weak form of the governing equations to a local subdomain around each node, and uses a set of shape functions (or basis functions) to approximate the unknown variables. The shape functions are constructed using the moving least squares (MLS) technique, which fits a polynomial function to the nodal values in the local subdomain. The advantage of MLPG is that it can handle complex geometries, moving boundaries, large deformations, and nonlinear problems with high accuracy and efficiency.

2 How does MLPG work?

MLPG consists of four main steps: node distribution, shape function construction, local integration, and global assembly. First, the nodes are distributed over the fluid domain, either randomly or according to some criteria such as density, gradient, or curvature. The nodes can be fixed or moving, depending on the problem. Second, the shape functions are constructed using the MLS technique, which requires a weight function to determine the influence of each node on the local subdomain. The weight function can be chosen based on the distance, smoothness, or compactness criteria. Third, the local integration is performed by applying the weak form of the governing equations to each local subdomain, and obtaining a system of equations for each node. The local integration can be done analytically or numerically, using different types of quadrature rules. Fourth, the global assembly is done by assembling the local systems into a global system, and solving it using appropriate solvers.

3 What are the advantages of MLPG?

MLPG offers several advantages over FEM and other methods for fluid dynamics. It does not require mesh generation or refinement, which can save time and prevent errors. MLPG can handle complex geometries and irregular boundaries, as well as moving boundaries and large deformations without any remeshing or reconnection. Moreover, it is capable of dealing with problems involving fluid-structure interaction, multiphase flow, and free surface flow. MLPG can achieve high accuracy and stability by using higher-order shape functions and adaptive weight functions, while reducing numerical dispersion and diffusion with local conservation principles and special boundary treatments. Additionally, it reduces the computational cost and memory requirement by using a sparse global matrix and a small local subdomain, plus it can take advantage of parallel computing and GPU acceleration.

4 What are the challenges of MLPG?

MLPG is not a perfect method and has some challenges and limitations. For instance, it requires a sufficient number and distribution of nodes to ensure convergence and accuracy, as well as the right choice of weight function and shape function order for stability and smoothness. It can also suffer from ill-conditioning and singularity of the global matrix, numerical integration errors, boundary integration errors, difficulty in imposing boundary conditions, and coupling with other methods. Additionally, issues may arise with enforcing incompressibility and continuity for fluid problems.

5 How can MLPG be improved?

MLPG is a promising and powerful method for fluid dynamics, but further development and improvement is necessary. Possible ways to improve MLPG include developing new and better techniques for node distribution and placement, weight function and shape function selection, local and global integration, as well as boundary condition imposition and method coupling. For example, adaptive node refinement, variable shape function order, higher-order quadrature rules, penalty methods, and domain decomposition methods can all be used to enhance MLPG.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?