Unsplit Semi-Lagrangian CIP Fluid Solver for Stable Advection
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Solution Overview
Problem
Existing fluid simulation methods face challenges in achieving accurate and stable advection without dimensional splitting, leading to numerical dissipation and diffusion, especially when dealing with complex boundaries and adaptive grids.
Innovation Solution
A semi-Lagrangian CIP fluid solver is developed without dimensional splitting, using an unsplit semi-Lagrangian constrained interpolation profile (USCIP) method that utilizes all derivative information and additional polynomial terms to improve stability and accuracy, while maintaining computational efficiency.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Stability of the object's composition
If dimensional splitting is used in semi-Lagrangian CIP method, then stability is improved, but computational load increases and numerical dissipation occurs
Solution Approach 1:
The patent applies segmentation by splitting the advection process into multiple sub-steps, where each sub-step uses a simplified CIP method with reduced computational complexity. This allows the method to maintain stability through multiple smaller computational stages while avoiding the full computational burden of traditional dimensional splitting approaches.
Solution Approach 2:
The patent changes the parameter of the CIP method by modifying the polynomial order and selection criteria for different spatial dimensions. By adaptively selecting polynomial degrees based on local flow characteristics and grid resolution, the method achieves stability without requiring the heavy computational resources associated with full dimensional splitting.
2Stability of the object's composition
If dimensional splitting is used in semi-Lagrangian CIP method, then stability is improved, but numerical diffusion increases
Solution Approach 1:
The patent applies dynamics by making the CIP method adaptive to local flow conditions. The polynomial order and interpolation scheme are dynamically selected based on local velocity gradients, grid spacing, and flow direction, allowing the method to maintain high accuracy and minimize numerical diffusion while achieving stability in varying flow regimes.
Solution Approach 2:
The patent applies local quality by using different polynomial orders and CIP formulations for different spatial locations and flow conditions. In regions with smooth flow, lower-order polynomials are used to reduce computational cost, while in regions with sharp gradients or complex flow patterns, higher-order polynomials are employed to maintain accuracy and reduce numerical diffusion.
3Measurement precision
If wide stencils are used in ENO/WENO methods, then advection accuracy is improved, but handling of complex boundaries becomes problematic
Solution Approach 1:
The patent applies local quality by adapting the stencil size and polynomial order to local flow conditions and boundary proximity. Near complex boundaries, the method automatically reduces the stencil width and polynomial order to accommodate geometric constraints, while in open domains with smooth flow, wider stencils and higher-order polynomials are used to maximize advection accuracy.
4Measurement precision
If higher-order CIP methods are used, then advection accuracy is improved, but computational complexity increases
Solution Approach 1:
The patent applies partial action by using higher-order CIP methods selectively only where needed. The method evaluates local flow characteristics and applies high-order polynomials only in regions with significant velocity gradients or complex flow patterns, while using lower-order methods in regions with smooth, simple flow to reduce overall computational complexity.
Data Source
AI summary
A new constrained interpolation profile method, which is stable and accurate but requires less amount of computation, is provided. CIP is a high-order fluid advection solver that can reproduce rich details of fluids. It has third-order accuracy but its computation is performed over a compact stencil. A novel modification of the original CIP method that fixes all of the above problems without increasing the computational load or reducing the accuracy is provided. The proposed method brings significant improvements in both accuracy and speed.