Hexahedral Elements Reducing Shear Locking via Aspect-Ratio Scaling
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Solution Overview
Problem
Fully-integrated 8-node hexahedral elements suffer from shear locking effects, particularly pronounced in elements with poor aspect ratios, leading to artificial stiffness and inaccurate structural simulations in finite element methods, which complicates the analysis of engineering products like cars and airplanes.
Innovation Solution
Introducing aspect-ratio based scale factors to modify partial derivatives of the isoparametric shape function and directly modifying off-diagonal components in the local Jacobian matrix to alleviate spurious shear deformation modes, effectively reducing artificial transverse shear locking effects.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If full integration with 8 Gauss-Legendre integration points is used, then all possible modes of deformation generate stress in the element, but shear locking effect occurs leading to artificial stiffness
Solution Approach 1:
The patent modifies the Jacobian matrix by scaling its components based on the element's aspect ratio. Specifically, the off-diagonal components are scaled by a factor that depends on the ratio of element dimensions, which adjusts the stiffness calculation to account for the element's geometry and reduces the artificial shear locking effect while maintaining accurate stress calculation
2Productivity
If elements with poor aspect ratio are used, then computational efficiency improves and model creation is easier, but shear locking effect is amplified
Solution Approach 1:
The patent introduces aspect ratio-based scaling factors that modify the Jacobian matrix components. The scaling factor is derived from the ratio of element dimensions (e.g., length to width), and when applied to the Jacobian matrix, it compensates for the geometric distortion in poor aspect ratio elements, thereby reducing shear locking and improving simulation accuracy without sacrificing computational efficiency
3Ease of manufacture
If 8-node hexahedral element is used, then model creation is easier and computational efficiency improves, but transverse shear locking occurs in pure bending
Solution Approach 1:
The patent applies scaling factors to the Jacobian matrix that are specifically designed to counteract the transverse shear locking effect in 8-node hexahedral elements. The scaling modifies the relationship between nodal displacements and element deformation, allowing the element to accurately represent pure bending behavior without developing artificial shear stiffness
Data Source
AI summary
Improved 8-node hexahedral elements configured for reducing shear locking in finite element method are disclosed. According to one aspect, aspect-ratio based scale factors are introduced to modify partial derivatives of the isoparametric shape function of the hexahedral element with respect to isoparametric dimensions, respectively. The modified derivatives are used for calculating the Jacobian matrix thereby the rate-of-strain. The scale factor is configured such that no changes for a perfect cubic solid element (i.e., element having aspect ratio of 1 (one) in all three spatial dimensions), while significant changes for element having poor aspect ratio. In other words, elements with poor aspect ratio are mapped to a perfect cubic element using the aspect-ratio based scale factors. According to anther aspect, off-diagonal components in the local Jacobian matrix are directly modified by cancelling terms related to spurious shear deformation modes. This measure completely alleviates the shear locking effect even for perfectly shaped elements.