LONG ARITHMETIC IN EXCEL. III. SOLUTION OF SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS FOR TEST PROBLEMS OF THE FINITE ELEMENT MODELLING

Abstract

The class of rational numbers and full precision algorithms are adopted to solve systems of the linear algebraic equations. The simple method of Gaussian elimination is implemented which shows typical computational procedures and costs of computer resources, especially the time of calculus.
One practical application of this algorithm is test tasks solving and software verifying. For example, the finite element method is often used in the solid mechanics modelling. Brief theoretical basis of the variational problem of bending of a complaint to shear plate under constant pressure is given. The perspective scheme of the FEM using B-splines is proposed.
For example, one-dimensional finite elements using quadratic opened B-splines as basis functions are considered. Typical matrices contain the product of the basis functions and their first derivatives in various combinations under the integral. In the case of a single length of these elements, the matrices are obtained analytically in the form of rational numbers. The assembly procedure for ten elements with two degrees of freedom on each is illustrated. The sparse global matrix and right handle size of the resulting system with rational coefficients are constructed and its exact solution is obtained.
Also examples of filled symmetric matrices are considered. A case of rigid systems with ill-conditioned Gilbert's matrices dimensionality up to N=200 is examined. The maximum number of digits of a denominator is 240 for N=200, what explains the inaccessibility of exact solutions when using double precision.
Numeric simulation confirmed the expected analytical precision and the robustness of the proposed program code. In particular, the time cost and increasing of rational coefficients size during the direct way of the Gauss method are traced. We compared a "slow" and "fast" computers with 2 processors and 2 MB of RAM vs. 4 processors and 8 MB of RAM respectively. The calculation time was reduced by three times. This indicates that significant computer resources are critically required. On the other hand, long arithmetic using binary algorithms can be implemented more efficiently.

Keywords: EXCEL Add-In, symbolic computations, systems of linear algebraic equations with rational coefficients, modelling, FEM, software testing and verifying.