B-Spline Reduced Order Models of the Multidimensional Fields ION LIXANDRU, VASILE MARINESCU, GABRIEL FRUMUŞANU, MARIAN VIOREL CRĂCIUN, ALEXANDRU EPUREANU Manufacturing Science and Engineering Departament „Dunărea De Jos” University of Galati 111 Domneasca , 800201 - Galati ROMANIA E-mail: Alexandru.Epureanu@ugal.ro Abstract: - There are many situations when, to solve a problem connected to machining processes, models of thermal or mechanical fields involved by the cutting process are required; because they are multidimensional fields, an appropriate mathematical tool to be used is required. The analytical approach of B-spline functions presents a number of disadvantages related to the great volume of calculations and, it also does not easily allow passing toward multivariable B-spline functions. The use of tensor fields, by means of the tensor product and the contraction products, enables a natural transition from B-spline functions of one variable to multivariable B-spline functions, acquiring some useful functional minimizations in practical applications. The assessment of the value of a multivariable B- spline function, which comes from the tensor product of B-spline functions of one variable, assumes the recurrence of the same calculation algorithm in each coordinate direction, for each B-spline function of one variable. The final result is the same, no matter the calculation order we choose, for different directions. In this paper we present methods for the assessment of B-spline functions for a given point, by means of tensor fields and methods for the selection of the initial data. On this basis, we developed different algorithms for testing the theoretical solutions we obtained. The assessment algorithm of B-spline function of one real variable, based on convex combinations, provides the numerical stability of the process. The assessment of a multivariable B-spline function is made by a two-steps algorithm, in each step being evaluated a B-spline function of one variable. In practice, the components of the vector or matrix of the control points are experimentally determined and, therefore, it is not known their real value, but an approximation of them. Considering the B-spline function depending on these values by minimizing operators as: dt t f 2 , dudv v u x 2 , , dvdw du w v u x 2 , , , out of the minimum necessary conditions for a multivariable function, we get homogenous linear algebraic systems, out of whose analysis we can obtain useful information on the importance of points at which measurements are made. Key-words: - B-spline function, tensorial fields, the assessment algorithms, minimizing the functional operators, the most important control points, the convex combinations. 1 Introduction In what follows we use EINSTEIN notation: whenever an index variable appears twice in a monomial, it implies we are summing over all of its possible values. We also note the tensor product with , and the contraction product of index i with i . The last one produces a new tensor whose parts are to be found by summing the components of the two tensors after the common index, i. In [5] and [6] the above operations are presented as it follows: 1) Given t m u u u u ,..., , 2 1 ; t n v v v v ,..., , 2 1 , then n h n k T T v u kh , 1 ; , 1 , where n h m k v u T h k kh , 1 ; , 1 , . 2) Given ; ,..., , 2 1 t n v v v v n k j jk T T , 1 , , then T v n k j i ijk U U , 1 , , , where n k j i T v U jk i ijk , 1 , , , . 3) Given n j i ij T T , 1 , ; n h k kh U U , 1 , . U T n h k j i ijkh W W , 1 , , , , where n h k j i U T W kh ij ijkh , 1 , , , , . 4) If t n u u u u ,..., , 2 1 ; t n v v v v ,..., , 2 1 then: u u v u v v t i i i , u v u v u v j j j . 5) Given n j i ij a A , 1 , ; t n v v v v ,..., , 2 1 v, then: A A Av v j ; A v v t i . Proceedings of the 11th WSEAS Int. Conf. on MATHEMATICAL METHODS, COMPUTATIONAL TECHNIQUES AND INTELLIGENT SYSTEMS ISSN: 1790-2769 224 ISBN: 978-960-474-094-9