Two-Level Mesh-Free Shape Function Transformation
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Solution Overview
Problem
Conventional finite element analysis (FEA) methods face challenges such as mesh distortion, high computational costs, and numerical errors in nonlinear and large deformation simulations, particularly in mesh-free methods, due to high-order integration rules and lack of interpolation properties.
Innovation Solution
A two-level transformation scheme is introduced to transform original mesh-free shape functions into second transformed shape functions, which filter out high-order terms using a low-pass filter, allowing for low-order integration and simplifying boundary condition treatments, thereby reducing computational resources.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If mesh-free methods are used to avoid mesh distortion and improve flexibility in nonlinear analysis, then the ability to handle large deformation and geometric non-linearity is improved, but the computational cost (CPU time) increases significantly
Solution Approach 1:
The patent changes the mathematical parameters of the shape functions by introducing a transformation that modifies the original mesh-free shape functions into transformed versions with different properties. This transformation alters the integration requirements from high-order to low-order, thereby reducing computational cost while preserving the ability to handle large deformations and geometric non-linearity
2Measurement precision
If high-order integration rules are used in traditional mesh-free methods to maintain accuracy, then the precision of spatial integration is improved, but the computational complexity and CPU cost increase
Solution Approach 1:
The patent transforms the shape functions such that their mathematical properties change from requiring high-order integration to allowing low-order integration. The transformed shape functions maintain sufficient accuracy for spatial integration while dramatically reducing the computational complexity and number of integration points required
Solution Approach 2:
The patent effectively replaces expensive high-order integration rules with cheaper low-order integration rules. The transformed shape functions are designed to be computationally inexpensive to integrate, achieving the desired accuracy with significantly less computational resources
3Device complexity
If original mesh-free shape functions are used directly, then the method simplicity is maintained, but the imposition of essential boundary conditions becomes difficult and the convergence rate decreases
Solution Approach 1:
The patent performs a preliminary transformation of the shape functions before the actual finite element analysis is conducted. This pre-transformation ensures that the shape functions possess the necessary properties (Kronecker delta properties) to easily impose essential boundary conditions and achieve good convergence rates, avoiding the need for complex boundary condition handling during the analysis
4Measurement precision
If a fine mesh is used in FEA to accurately capture high gradients or local characteristics, then the accuracy in problems with high gradients is improved, but the computational cost increases dramatically
Solution Approach 1:
The patent divides the continuum into discrete nodal points without requiring a connected mesh structure. This segmentation allows the method to capture local characteristics and high gradients through the distribution and influence domains of individual nodes, rather than relying on fine mesh elements, thereby reducing computational cost while maintaining accuracy
Data Source
AI summary
A two-level transformation scheme to enable a practical fast mesh-free method is disclosed. The first level transformation transforms the original chosen mesh-free shape function to a first transformed mesh-free shape function that preserves Kronecker delta properties. The first transformed mesh-free function allows the essential boundary conditions to be imposed directly. The second-level transformation scheme employs a low pass filter function served as a regularization process that filters out the higher-order terms in the monomial mesh-free approximation obtained from the first-level transformation scheme with desired consistency and completeness conditions. This integration scheme requires only a low-order integration rule comparing to the high order integration rule used in the traditional mesh-free methods. The present invention simplifies the boundary condition treatments and avoids the usage of high-order integration rule and therefore is more practical than the traditional mesh-free methods.