A Topological Extremum Solving Method for the Navier-Stokes Equations Based on the boldsymbol{Pi} Projection Operator and the MOC-DOG-ECS-MlE Framework

Author: Zhang Suhang

1. Introduction

The Navier-Stokes (NS) equations serve as the core governing system in viscous fluid dynamics. Traditional Computational Fluid Dynamics (CFD) has long been grounded in a global, single-origin Euclidean continuous space, relying on point-by-point differential discretization and iterative solving of partial differential equations. This paradigm inherently suffers from limitations: nonlinear convective terms easily induce numerical oscillations, low-order schemes are accompanied by pseudo-diffusion effects, pressure-velocity coupling remains weak, and long-term transient computations often exhibit conservation drift. Furthermore, at the theoretical level, it struggles to explain the gradient evolution of boundary layers and the global smoothness of flow fields.

Departing from traditional differential solution paths, this paper introduces the boldsymbol{Pi} projection operator as a spatial transformation tool. Relying on the complete geometric dynamic framework of MOC-DOG-ECS-MlE, the continuous NS equations are transformed into a constrained functional extremum problem over a discrete topological domain. This establishes a top-down topological evolution solving paradigm that balances the stability of CFD engineering computations with the theoretical analysis of NS regularity.

2. Traditional NS Governing Equations and Inherent Limitations

The standard incompressible viscous NS equations are:

begin{cases}

dfrac{partial boldsymbol{u}}{partial t} + (boldsymbol{u}cdotnabla)boldsymbol{u} = -dfrac{1}{rho}nabla p + nunabla^2boldsymbol{u}+boldsymbol{f}\

nablacdotboldsymbol{u}=0

end{cases}

Traditional finite difference and finite volume solving models present four main shortcomings:

1. Convective terms rely on local gradient approximations, making truncation errors unavoidable;

2. The global single-origin geometric assumption fails to adapt to local spatial distortions in boundary layers and shear layers;

3. Mass and momentum conservation depend on iterative relaxation approximations, constituting weak constraints;

4. Convergence relies on residual thresholds, lacking physical objectives based on natural evolutionary laws.

3. Logical Overview of the Overall Architecture

The overall system consists of a two-layer structure, maintaining unified terminology throughout the text:

1. Pre-requisite Tool: The boldsymbol{Pi} projection operator, responsible for mapping the continuous Euclidean fluid domain onto a discrete topological space;

2. Core Intrinsic Framework: MOC-DOG-ECS-MlE, which sequentially completes the construction of the geometric basis, the establishment of flow order, symmetric conservation constraints, and energy-optimal evolution.

boldsymbol{Pi} serves as an independent projection mapping tool, while the latter four constitute a closed-loop dynamic geometric framework.

4. Step-by-Step Mathematical Derivation and Physical Interpretation

4.1 The boldsymbol{Pi} Projection Operator: Mapping Continuous Space to Discrete Topological Domains

Phase-space dimensionality reduction projection is achieved using boldsymbol{Pi}:

boldsymbol{Pi}: Omega(boldsymbol{x}) mapsto Omega_d(boldsymbol{xi})

Here, boldsymbol{x} represents traditional global Euclidean coordinates, and boldsymbol{xi} denotes local coordinates in the topological domain. 

Effect: This decouples the flow field from its binding to a global fixed coordinate system, retaining only the topological connectivity between nodes, thereby laying the foundation for the Multi-Origin Coordinate (MOC) geometric reconstruction.

4.2 MOC (Multi-Origin Coordinates) High-Dimensional Geometry: Local Spatial Reconstruction

Within the projected discrete domain Omega_d, the concept of a single global origin is abandoned. Instead, independent local origins and local metric tensors are established for each fluid topological element:

ds_i^2 = g_{ab}^{(i)}dxi^a dxi^b

Rather than forcing the entire flow field to share flat Euclidean geometry, this allows for locally curved geometries near shear layers and walls, explaining velocity distortions in boundary layers.

4.3 DOG (Discrete Order Geometry): Topological Algebraic Reconstruction of Convective Terms

Based on the MOC local geometric basis, DOG constructs directed topological connections according to flow causality. It equivalently transforms the difficult-to-handle nonlinear convective differential term into a weighted algebraic summation of upstream and downstream nodes:

left[(boldsymbol{u}cdotnabla)boldsymbol{u}right]i triangleq sum{jinmathcal{N}(i)} w_{ij}(boldsymbol{u}_j-boldsymbol{u}_i)

w_{ij} represents the DOG flow order weights, strictly following the upwind transport direction. This eliminates the nonlinearity of the convective term, fundamentally suppressing numerical oscillations and pseudo-diffusion.

4.4 ECS (Equilibrium Constraint System): Symmetric Balance Stability Constraints

Based on the principle of least action and spatiotemporal symmetry conservation, the differential form of the continuity equation is upgraded to a hard balance constraint of nodal fluxes:

sum_{jinmathcal{N}(i)} F_{ij}=0,quad delta S = 0

ECS simultaneously carries the triple roles of flux conservation, spatiotemporal symmetry, and boundary steady-state restrictions. As a strong constraint, it does not rely on iterative relaxation corrections, avoiding mass drift in long-term simulations.

4.5 MlE (Minimal Level Efficiency): Global Evolution Convergence Criterion

A hierarchical dissipation functional is constructed to distinguish macroscopic kinetic energy from microscopic viscous shear dissipation:

mathcal{E}(boldsymbol{u},p)=int_{Omega_d}left( dfrac{1}{2}rho|boldsymbol{u}|^2+muPhi(nablaboldsymbol{u}) right)dOmega_d

Flow field iterative updates no longer rely on artificial residual thresholds; instead, they use functional minimization as the natural evolution objective:

boldsymbol{u}^{n+1}=boldsymbol{u}^n-alphanabla_{boldsymbol{u}}mathcal{E}

Non-physical solutions such as high-energy spurious vortices and local velocity spikes are automatically eliminated by the energy screening mechanism, preserving only low-dissipation steady states that conform to physical laws.

5. Unified Core Formula of the System

Integrating the projection mapping and the constraints of the entire framework, the final paradigm for topological NS solving is given by:

boldsymbol{U}^{n+1} = mathop{argmin}_{boldsymbol{U}inboldsymbol{prod}(Omega)} Big{ mathcal{E}(boldsymbol{U}) ,big|, text{MOC geometric basis},,text{DOG flow order},,text{ECS symmetric constraints} Big}

6. Regularity-Assisted Boundary Conditions
To prevent blow-up singularities in the wall-normal gradients, additional global prior asymptotic constraints are appended. These do not belong to the core framework but serve as supplementary boundary regularization:

\begin{cases}
\lim\limits_{t\to\infty}\boldsymbol{u}{\parallel}(t)=\boldsymbol{u}{\parallel}^*\in{\text{elliptic steady flow fields}}\
\exists C>0,\sup\limits_t|\nabla\boldsymbol{u}{\perp}(t)|{L^\infty(\partial\Omega)}\le C
\end{cases}

7. Chapter Summary
Using \boldsymbol{\Pi} as a pre-requisite spatial transformation tool and relying on the closed-loop MOC-DOG-ECS-MlE geometric dynamic framework, this paper accomplishes the paradigm shift of the NS equations from continuous partial differential solving to discrete topological constrained extremum problems. On one hand, this system enhances the convergence stability of CFD engineering simulations; on the other hand, it provides novel insights from the perspectives of geometry, symmetry, and dissipative evolution for studying the global smoothness and singularity regularity of the NS equations.