Lagrange Multiplier Correction in Modal Dynamic Analysis
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Solution Overview
Problem
Existing modal dynamic analysis methods using finite element models (FEMs) with Lagrange multipliers often produce grossly incorrect stress and reaction force results, making it difficult to achieve accurate results, even with increased eigenmodes.
Innovation Solution
The method involves calculating a correction term by solving a system of linear algebraic equations with a sparse matrix and modifying the modal analysis to alter Lagrange multipliers, allowing for accurate modeling of stress and reaction forces as a superposition of eigenmodes.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If modal dynamic analysis is performed using FEM with Lagrange multipliers, then the structural response can be modeled as a superposition of eigenmodes, but the stress and reaction force results become grossly incorrect
Solution Approach 1:
The patent introduces a correction term as an intermediary element that mediates between the erroneous Lagrange multipliers obtained from standard modal analysis and the accurate stress/reaction force calculations. This correction term is computed by solving a separate system of linear algebraic equations and then applied to recover the accurate Lagrange multipliers, thereby resolving the contradiction between computational efficiency and result accuracy.
Solution Approach 2:
The patent modifies the Lagrange multiplier parameters by adding a correction term to the originally computed values. This parameter change transforms the grossly incorrect Lagrange multipliers into accurate ones, enabling reliable stress and reaction force calculations while maintaining the efficiency of modal analysis methodology.
2Measurement precision
If the number of eigenmodes is increased to improve accuracy, then more structural response details are captured, but computational complexity increases
Solution Approach 1:
The patent segments the computational process into two distinct parts: (1) the standard modal analysis that computes eigenmodes and approximate Lagrange multipliers, and (2) a correction step that solves a separate system of linear algebraic equations. This segmentation allows the use of fewer eigenmodes in the first stage while recovering accuracy in the second stage, thereby reducing overall computational complexity compared to using many eigenmodes throughout.
Solution Approach 2:
The patent performs preliminary computation of eigenmodes and initial Lagrange multipliers using a reduced set of modes, then applies a correction term in a subsequent step. This preliminary action with fewer modes reduces computational complexity while the correction step restores accuracy, avoiding the need to compute and store many eigenmodes.
3Measurement precision
If a correction term is calculated and applied to alter Lagrange multipliers, then accuracy of stress and reaction forces is improved, but additional computational steps are required
Solution Approach 1:
The patent replaces the need for complex iterative mechanical equilibrium iterations with a direct solution of a system of linear algebraic equations. The correction term is obtained by solving (K + G^T * R * G) * dλ = -G^T * r, where K is the stiffness matrix, G is the constraint matrix, and r is the residual vector. This substitution of the computational approach maintains efficiency while improving accuracy.
Data Source
AI summary
Modal dynamic analysis for finite element models (FEMs) that include Lagrange multipliers may generate incorrect stress and reaction forces. Computer systems and computer-implemented methods are provided for modifying the modal analysis to correctly generate stress and reaction forces. The systems and methods perform the modal analysis by employing a FEM and modeling stress and reaction forces of the FEM using Lagrange multipliers. The systems and methods calculate a correction term that comprises corrected values of the Lagrange multipliers. The methods and systems modify (and improve) the modal analysis by using the correction term to correct the Lagrange multipliers of the FEM, which enables the modal analysis to generate correct stress and reaction forces.