Abstract
For constructing physical science based models in irregular numerical grids, an easy-to-implement method for solving partial differential equations has been developed and its accuracy has been evaluated by comparison to analytical solutions that are available for simple initial and boundary conditions. The method is based on approximating the local average gradients of a field by fitting equation of plane to the field quantities at neighbouring grid positions and then calculating an estimate for the local average gradient from the plane equations. The results, comparison to analytical solutions, and accuracy are presented for 2-dimensional cases.
Highlights
Aim of the present study is to describe a numerical method for solving partial differential equations (PDE’s) in deformed grids, which is relatively easy to implement
The method is designed for simulating the physical phenomena which occur at microstructural level during thermomechanical processing of materials, such as diffusional [1] and displacive phase transformations [2, 3, 4], elasticity and plasticity [5], recovery, recrystallization and grain growth [6, 7, 8], but it could be used in other contexts where PDEs need to be solved in irregular numerical grids
As an important step towards physics based full field modelling the microstructure evolution during thermo-mechanical processing, a numerical method which is relatively easy to implement and which can be used for solving partial differential equations in the deformed grids has been developed and its accuracy has been assessed by comparison to known analytical solutions
Summary
Aim of the present study is to describe a numerical method for solving partial differential equations (PDE’s) in deformed grids, which is relatively easy to implement. At the grid point where the local maxima/minima appears in the numerical simulation, the gradient vector quantities were calculated using Eq (7) at both the points denoted with red filled circles, as well as at the points denoted with green unfilled circles in Fig. 3 instead of using the previously calculated gradient at the points pn, pe, ps, pw. These values were used when calculating the second order derivatives to obtain the correction which helped to avoid the oscillations observed in the simulations with sparse grids
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