A proper orthogonal decomposition variational multiscale meshless interpolating element‐free Galerkin method for incompressible magnetohydrodynamics flow

Abstract

Summary In the recent decade, the meshless methods have been handled for solving most of PDEs due to easiness of the meshless methods. One of the popular meshless methods is the element‐free Galerkin (EFG) method that was first proposed for solving some problems in the solid mechanics. The test and trial functions of the EFG are based on the special basis. Recently, some modifications have been developed to improve the EFG method. One of these improvements is the variational multiscale EFG procedure. In the current article, the shape functions of interpolation moving least squares approximation have been applied to the variational multiscale EFG technique for solving the incompressible magnetohydrodynamics flow. In order to reduce the elapsed CPU time of simulation, we employ a reduced‐order model based on the proper orthogonal decomposition technique. The current combination can be referred to as the reduced‐order variational multiscale EFG technique. To illustrate the reduction in CPU time used as well as the efficiency of the proposed method, we applied it for the two‐dimensional cases.

An adaptive reduced order model for the angular discretization of the Boltzmann transport equation using independent basis sets over a partitioning of the space‐angle domain

“…The RDPOD basis sets for each Z q,r can now be formed through the SVD of each snapshot matrix, S q,r = U q,r Σ q,r V T q,r , (10) where U q,r and V q,r are unitary matrices of sizes N a ∕8 × N a ∕8 and n s × n s , respectively. The column vectors of U q,r contain the optimized basis vectors that best represent the snapshot data, ordered such that the first n a columns form the optimal n a basis vectors in the Frobenius norm.…”

“…ROMs have been applied extensively across many fields of engineering. Recent developments include the application of proper orthogonal decomposition (POD) to transient heat conduction, 7 turbulent supersonic jets, 8 acoustic waves, 9 incompressible magnetohydrodynamics, 10 and many other problems. POD‐based methods have been combined with neural networks to model plasticity, 11 unsteady flows in a combustion problem, 12 the viscous Burgers equation, 13 and structural damage 14 .…”

A Dimension Splitting Generalized Interpolating Element-Free Galerkin Method for the Singularly Perturbed Steady Convection–Diffusion–Reaction Problems

“…By coupling the variational multiscale method and EFG method, Ouyang et al established the variational multiscale EFG method (VMEFG) [25]. The VMEFG method has been successfully applied in many physical problems [26][27][28]. However, the disadvantage of VMEFG is that the bubble function in the stabilization factor must be based on the mesh and is difficult to construct in high-dimensional cases.…”