Orthogonal Basis Bubble Function Element Numerical Analysis Method
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Solution Overview
Problem
Conventional bubble function elements in finite element analysis require large memory and computation time for inverse matrix calculations, leading to high manufacturing costs and delayed analytical processes, while approximated lumped mass matrices result in low precision and reliability of analysis results.
Innovation Solution
The orthogonal basis bubble function element method involves acquiring a consistent mass matrix, diagonalizing it to generate a diagonal mass matrix for each element, and computing its inverse for precise analysis without approximations, ensuring high reliability and efficiency.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If consistent mass matrix is used for bubble function element analysis, then analysis precision is improved, but memory requirements and computation time increase significantly
Solution Approach 1:
The consistent mass matrix is segmented and diagonalized element-by-element. Each element's mass matrix is processed independently to generate a diagonal approximation, avoiding the need to compute and store the inverse of the full global consistent mass matrix. This segmentation reduces memory requirements and computation time while maintaining analysis precision.
Solution Approach 2:
The mass matrix representation is changed from a full dense matrix to a diagonal matrix through the diagonalization process. This parameter change transforms the storage structure from O(n²) to O(n), significantly reducing memory requirements while preserving the essential mass distribution characteristics needed for accurate analysis.
2Measurement precision
If consistent mass matrix is used for bubble function element analysis, then analysis precision is improved, but manufacturing cost increases
Solution Approach 1:
By segmenting the mass matrix computation to the element level and applying diagonalization, the computational burden is distributed across multiple small, independent operations. This reduces the need for high-performance computing hardware and large memory capacities, thereby lowering manufacturing costs of the computational system.
Solution Approach 2:
Transforming the mass matrix to diagonal form changes the computational parameters from requiring large memory capacity and high processing power to requiring minimal resources, making the system more cost-effective to manufacture and deploy.
3Productivity
If lumped mass matrix approximation is used, then computation time is reduced, but analysis reliability decreases
Solution Approach 1:
The invention changes the approximation parameter from a fully lumped mass matrix (summing all off-diagonal terms to diagonal) to a diagonaled consistent mass matrix (element-by-element diagonalization). This parameter change maintains better physical accuracy while achieving similar computational efficiency, thereby improving analysis reliability without sacrificing productivity.
Data Source
AI summary
A known analytical physical quantity of an analysis subject and an element level consistent mass matrix of each element are obtained by acquiring units. A bubble function is integrated for each element, and the element level diagonal mass matrix of each element is computed by substituting the value that is integrated for the element level consistent mass matrix of each element. The diagonal mass matrix for the entire analysis region is computed by the summation (superposition) of the element level diagonal mass matrices of each element and inverse matrix thereof is computed. A motion of the analysis subject is analyzed based on the known analytical physical quantity, the diagonal mass matrix for the entire analysis region, and the inverse matrix thereof.