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Computational Numerical Methods
Published in Timothy Bower, ®, 2023
Matlab has a function for computing definite integrals called integral, which is a replacement for an older function called quad. The integral function uses a global adaptive quadrature algorithm where the integration region is divided into piecewise smooth regions before integrating each subinterval. The widths of the subintervals are adapted so that each subinterval is accurately modeled by an equation.
Experimental and analytical evaluation of tool path error using computer integrated nonlinear kinematical modeling for a 4DOF parallel milling machine
Published in International Journal of Computer Integrated Manufacturing, 2021
Sina Akhbari, Mehran Mahboubkhah, Davoud Karimi, Ahmad Barari
However, the approximation accuracy of the numerical result calculated by the Cavalieri-Simpson formula directly cannot be guaranteed. To ensure the accuracy of the kinematic error calculation, it is useful to utilize the adaptive quadrature method which adjusts the step size to be smaller over portions of the curve where large functional variation occurs (Mathews and Fink 2004). In this method an adaptive bisection technique which can keep the result under a specific tolerance of its true value is combined with Cavalieri-Simpson method. The initial intervalis bisected into two equal subintervals and, and applying Eq. (34) recursively over each subinterval, only two additional evaluations ofare needed, shown in Eq. (35):
On measurement of dynamic modulus for bituminous mixtures
Published in International Journal of Pavement Engineering, 2019
S. Deepa, U. Saravanan, J. Murali Krishnan
In the first approach, the stress history used for computation of strains consists only the 10 test cycles for a specific frequency. As the first step, the initial strain value at the start of each frequency is zeroed. The next task is to choose the material parameters for the linear viscoelastic model for the 10 test cycles such that the error between computed and measured strain is minimised. Thus for each frequency, the 10 test cycle data consisting of 500 data points are used for modelling. The simulations are carried out using MATLAB (2010). Since the seating load is very less compared to the load associated with the test cycles, this is neglected in the simulations. Integration is carried out using ‘quadgk’ function in MATLAB (2010) which uses high-order global adaptive quadrature method. The model parameters are estimated using ‘lsqcurvefit’ tool which is a nonlinear least-squares solver in MATLAB (2010).