Dimensional Analysis in Mathematical Modeling Systems: A Simple Numerical Method
Orsa Journal On Computing
This paper develops a numerical method for dimensional analysis, which has previously been treated as a problem requiring symbolic mathematics techniques. The numerical method obeys the laws of dimensional arithmetic. This is achieved by specifying an encoding of units of measurement as prime numbers, and manipulating the resulting expressions numerically. The unique factorization theorem is applied to show that this method makes trivial the problems of dimensional simplification and
... n of dimensional equivalency, which are central issues in dimensional arithmetic. The solution has applications in mathematical modeling systems, chiefly in the model validation and model solution phases. The paper describes a first-order logic language, Ldim, which can be used in such systems to represent and manipulate dimensional information. This paper presents an efficient and simple numerical method for dimensional analysis, and discusses its applications in mathematical modeling systems. The problem of dimensional manipulation can be viewed as one of symbolic mathematics [25, 26] , since dimensions (quantities, units of measurement, see §2) are non-numeric symbols. However, we transform it to a simple numerical problem, and develop an algorithm for this transformed problem. There are three key steps in this approach. First, we recognize the special nature of the laws of dimensional arithmetic. Second, we develop a prime-encoding of dimensions, in which each unit of measurement is represented by a prime number. Third, we apply the unique factorization theorem from number theory to show that numeric arithmetic applied to this prime-encoding obeys the laws of dimensional arithmetic. Dimensional Analysis and Modeling Systems Dimensional arithmetic, or the calculus of dimensions, involves operations on dimensions analogous to the arithmetic operations on numbers. The techniques required to perform dimensional arithmetic as a symbolic mathematics problem are implemented in several computer algebra programs (e.g., Macsyma , Reduce , and Mathematica ). These systems symbolically solve problems such as proving that (a 2 + ab + b 2 + ba) = (a + b) 2 . With some effort, since the laws of physical algebra are a minor variant on those of standard arithmetic performed on numbers , dimensional arithmetic can be, and has been, performed using these systems. Our alternative, numerical, method does not require specialized symbolic manipulation techniques, and has been used in implementing features for dimensional analysis in a model management system TEFA, described in  .