Cell-Centered Finite Difference Method for Unstructured Grid Fluid Simulation
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Solution Overview
Problem
Current computer modeling methods for fluid dynamics, such as Traditional Finite Difference Methodology, are limited by their requirement for structured grids and struggle with complex geometries, leading to inaccuracies and convergence issues when dealing with compressible and non-compressible fluids, especially in irregular domains.
Innovation Solution
The development of a Cell-Centered Finite Difference Method (CCFDM) that allows for the use of arbitrary mesh topologies, enabling more flexible and accurate modeling of fluid flows by discretizing partial differential equations directly in physical space without the need for coordinate transformations, thus combining the efficiency of FDM with the flexibility of FVM and FEM.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If Traditional Finite Difference Methodology is used, then computational efficiency is improved, but the method is limited to structured grids and cannot handle complex geometries
Solution Approach 1:
The patent segments the computational domain into unstructured elements (triangles in 2D, tetrahedra in 3D) that can be assembled to form complex geometries. This allows the method to handle irregular domains while maintaining computational efficiency through element-based discretization, resolving the contradiction between structured grid requirements and geometry flexibility.
Solution Approach 2:
The patent introduces a dual mesh approach where a primal mesh represents the physical domain geometry and a dual mesh represents the computational structure. This dimensional transformation allows traditional finite difference methods to operate on unstructured grids by mapping problems between these two mesh types, achieving both geometric flexibility and computational efficiency.
2Adaptability or versatility
If coordinate transformations are applied to map physical domain to computational domain, then curvilinear domains can be handled, but numerical discontinuities in transformation metrics cause accuracy and convergence issues
Solution Approach 1:
The patent extracts and eliminates the coordinate transformation step from the traditional finite difference methodology. By using unstructured grids directly in the physical domain, the method removes the source of numerical discontinuities in transformation metrics, thereby maintaining solution accuracy and convergence while still being able to handle curvilinear and complex geometries.
Solution Approach 2:
The patent introduces an intermediary dual mesh structure that mediates between the physical unstructured mesh and the computational requirements. This dual mesh approach allows flux calculations and boundary conditions to be evaluated accurately without requiring coordinate transformations, thus avoiding the accuracy and convergence problems associated with transformation metrics.
3Adaptability or versatility
If Cell-Centered Finite Volume Methodology is used, then grid arrangement flexibility is improved, but inaccuracies in flux calculation across cell faces occur
Solution Approach 1:
The patent merges the advantages of cell-centered finite volume methodology (unstructured grid flexibility) with finite difference methodology (accurate flux calculations). By using a dual mesh approach where the dual mesh provides structured connectivity for accurate finite difference flux calculations, the method achieves both grid flexibility and flux calculation accuracy simultaneously.
Data Source
AI summary
A system and method for determining fluid flow of compressible and non-compressible liquids includes an input receiving an object model defined as a plurality of cells having nodes, a processor and memory. The processor is configured for: discretizing a partial differential equation (PDE) corresponding to the received model; for each node P: (i) locating all neighboring cells that share the node P; establish a finite difference stencil at each cell center, and identify stencil intersection points with cell boundary edges; calculate an approximate solution of the PDE at the intersection points; (ii) approximating the PDE at the cell center of the neighboring cells using the stencil and discretized PDE; and (iii) updating a solution of the PDE at the node P by using the solution of approximated discretized PDE at all the neighboring cell centers; and iteratively updating the solution for all nodes P from an initial guess until a convergence criterion is satisfied.