Mathematical Approximation
Numerical techniques facilitate the solution of differential and integral equations by requiring the approximate solution to satisfy the equation exactly at specific points. The collocation method utilizes these discrete points to determine the unknown coefficients of a chosen basis function. It finds frequent application in viscoelastic modeling where continuous time functions are converted into manageable algebraic forms.
Computational Execution
Choosing an appropriate set of points such as gauss legendre or chebyshev nodes improves the accuracy of the final result. Within the collocation method, the distance between these points directly influences the stability of the numerical inversion. A system of linear equations emerges from this process.
Error Distribution
Residual values at non collocated points indicate the quality of the approximation across the entire domain. Although the collocation method ensures zero error at the selected nodes, the deviation between nodes must be monitored to prevent oscillations. Precise selection of the nodal distribution minimizes these fluctuations which are sometimes called the runge phenomenon.
Accurate placement of nodes is especially important when dealing with sharp transitions in the material response.
Parameter Convergence
Integration into software allows for the rapid fitting of experimental data to complex constitutive models. The collocation method simplifies the transformation from the time domain to the frequency domain in polymer science. Proper execution leads to a stable representation of the relaxation modulus.
