Mathematical and Computational Sciences Division Projects


NIST Collaborative Projects


Building and Fire Research Laboratory

Mathematical Problems in Construction Metrology

The Building and Fire Research Laboratory (BFRL) at NIST is currently testing the effectiveness of LADAR (Laser Direction and Ranging) technology for modeling and locating equipment at construction sites. The Mathematical and Computational Sciences Division (MCSD) of the Information Technology Laboratory (ITL) at NIST has provided modeling and computational support to this effort in several areas. This support involved four major tasks.

Manufacturing Engineering Laboratory

Optimization Problems in Smart Machining Systems

Smart Machining Systems (SMS) aim at producing the first and every product correct; improving the response of the production system changes in demands; realizing rapid manufacturing; and, providing data on an as needed basis. An SMS is envisioned to contain a dynamic process optimizer with the capability of assessing the quality of the work and outputs of the SMS as well as improve itself over time. The dynamic optimizer builds and satisfies objective functions using machining models. It includes constraints from design such as dimensional and geometrical tolerances, surface integrity and surface quality. A general machining optimization problem consists in determining some decision variables, such as feed, depth of cut, spindle speed, in such a way that a set of constraints is satisfied and a desired objective function is optimized. F. Potra and D. Gilsinn are investigating the use of robust optimization methods. These methods aim at determining the decision variables such that the objective function is minimized and constraints are satisfied for all possible values of the machining parameter ranges. An objective function and constraints were defined for a preliminary SMS machining model. The objective function was solved by a linear program. The code was incorporated into a GUI for the machine operators by MEL collaborators and was used to machine a test part


Research and Development in Mathematical Methods and Software


Approximating Solutions of Delay Differential Equations

D. Gilsinn continues research on approximation of periodic solutions for delay differential equations (DDEs). This class of equations occurs in the stability analysis of machine tool vibrations, an area of metrology interest in the Manufacturing Engineering Laboratory (MEL) at NIST. A discrete Fourier representation algorithm has been developed that lends itself to efficient matrix-vector operations. The stability of solutions for DDEs with periodic coefficients is determined by certain eigenvalues that are called the characteristic multipliers. One method for developing characteristic multipliers for periodic solutions of delay equations involves solving an eigenvalue problem for an integral operator. The eigenvalues are estimated by those of an associated discretized linear operator. A pseudospectral collocation algorithm has been developed to approximate the fundamental matrix of the variational equation about the approximate solution. This algorithm has been included as an alternative to using a numerical delay solver for the fundamental matrix. In collaboration with F. Potra, an article has been submitted to the Journal of Integral Equations and Applications that analyses the convergence properties of the eigenvalues of the discretized integral operator to those of the integral operator.

Volume Estimation of Molded Industrial Artifacts by B-Splines

D. Gilsinn of ITL, B. Borchardt of MEL, and A. Tebbe of St. Mary's College of Maryland have investigated a method of estimating volumes of some semi-spherical molded artifacts produced for the Food and Drug Administration (FDA) by an industrial contractor. These artifacts are simulated lung cancer nodules, called phantoms, used to compare software on computed tomography (CT) scanners. MEL produced surface coordinate points on the molded artifacts by a coordinate measuring machine (CMM). This data was converted to spherical coordinates and modeled by tensor products of B-Splines. These were used since they have a small support set and lead to sparse matrices in the fitting process. Once B-Spline models were fit to the spherical coordinates the functions were used to extrapolate the data onto uniform grids on the surfaces that were then used along with the classic Divergence Theorem to estimate the artifacts' volumes. A bootstrap mehtod was used to estimate uncertainties. Straightforward least squares fits of the B-Spline models produced oscillations at the semi-sphere poles. Regularization methods are being studied to smooth the oscillations.

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Last change in this page : 11 September 2001.