Exact Boundary Derivative Formulation for Numerical Conformal Mapping Method

Conformal mapping is a useful technique for handling irregular geometries when applying the finite difference method to solve partial differential equations. When the mapping is from a hyperrectangular region onto a rectangular region, a specific length-to-width ratio of the rectangular region that fitted the Cauchy-Riemann equations must be satisfied. In this research, a numerical integral method is proposed to find the specific length-to-width ratio. It is conventional to employ the boundary integral method (BIEM) to perform the conformal mapping. However, due to the singularity produced by the BIEM in seeking the derivatives on the boundaries, the transformation Jacobian determinants on the boundaries have to be evaluated at inner points instead of directly on the boundaries. This approximation is a source of numerical error. In this study, the transformed rectangular property and the Cauchy-Riemann equations are successfully applied to derive reduced formulations of the derivatives on the boundaries for the BIEM. With these boundary derivative formulations, the Jacobian determinants can be evaluated directly on the boundaries. Furthermore, the results obtained are more accurate than those of the earlier mapping method

Simulation of 2D Free-surface Potential Flows Using a Robust Local Polynomial Collocation Method

Abstract In this paper a mesh-free numerical model for simulating 2D free-surface potential flows is established. A Lagrangian time-marching scheme is chosen for the boundary conditions of the moving and deforming free surface while a local polynomial collocation method is applied for solving the Laplace equation at each time step. The collocation method is developed in a way that the governing equation is satisfied on boundaries as well as boundary conditions. At any free surface node, this gives accurate estimation of the derivatives of velocity potential, which represent components of the velocity vector at that specific node. Therefore, trajectories of the free surface nodes can be predicted precisely. The numerical model is applied to the simulation of free surface waves in the liquid sloshing of a swaying tank. Present model is verified by comparing the numerical results with experimental data. Fairly good agreements are observed

Numerical Model Computing Wave Propagations in an Open Coast

Source: https://erdc-library.erdc.dren.mil/jspui/In this report, numerical models, based on the parabolic approximation for computing wave propagation, are presented. The theoretical background for the mild-slope assumption and the parabolic approximation is first summarized. Three different numerical models, using different coordinate systems, are then developed. These models calculate the combined wave refraction and diffraction over varying depth and current and in the vicinity of coastal structures. To verify the validity of the numerical models, numerical results are compared with laboratory and field data

A Review on the Modified Finite Point Method

The objective of this paper is to make a review on recent advancements of the modified finite point method, named MFPM hereafter. This MFPM method is developed for solving general partial differential equations. Benchmark examples of employing this method to solve Laplace, Poisson, convection-diffusion, Helmholtz, mild-slope, and extended mild-slope equations are verified and then illustrated in fluid flow problems. Application of MFPM to numerical generation of orthogonal grids, which is governed by Laplace equation, is also demonstrated