Question 1

What are eigenvalues and how are they used in CAE?

Accepted Answer

An eigenvalue λ satisfies Av=λv, where v is the corresponding eigenvector. In structural CAE, solving the generalized eigenvalue problem (K−λM)v=0 for stiffness matrix K and mass matrix M yields the natural frequencies and mode shapes of a structure. This is the foundation of FEM modal analysis.

Question 2

Does an inverse matrix exist when the determinant is zero?

Accepted Answer

When det(A)=0, matrix A is called singular and no inverse exists. In FEM, an improperly constrained stiffness matrix becomes singular (the structure has unconstrained rigid-body motion). Only when det(A)≠0 is the inverse defined as A⁻¹=(1/det(A))·adj(A).

Question 3

What is LU decomposition and how does it differ from Cramer's rule?

Accepted Answer

LU decomposition factors A=LU into lower and upper triangular matrices, allowing Ax=b to be solved efficiently in two stages: Ly=b (forward substitution) and Ux=y (back substitution). Computational cost is O(n³). Cramer's rule requires n+1 determinant evaluations and is impractical for n≥4. Large-scale CAE solvers use LU/Cholesky decomposition or iterative methods like conjugate gradient.

Question 4

How do matrix eigenvalues relate to principal stress directions?

Accepted Answer

The stress tensor σ (a symmetric matrix) has eigenvalues equal to the principal stresses σ₁, σ₂, σ₃, and eigenvectors give the principal stress directions. The Mohr's circle is a 2D visualization of this. For FEM post-processing, Von Mises stress is computed from principal stresses as σ_vm=√(½[(σ₁−σ₂)²+(σ₂−σ₃)²+(σ₃−σ₁)²]).

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