A Combined Integrating and Differentiating Matrix Formulation for Boundary Value Problems on Rectangular Domains

Integrating and differentiating matrices allow the numerical integration and differentiation of functions whose values are known at points of a discrete grid. Previous derivations of these matrices have been restricted to one-dimensional grids or to rectangular grids with uniform spacing in at least one direction. The present work develops integrating and differentiating matrices for grids with non-uniform spacing in both directions. The use of these matrices as operators to reformulate boundary value problems on rectangular domains as matrix problems for a finite-dimensional solution vector is considered. The method requires non-uniform grids which include "near-boundary" points. An eigenvalue problem for the transverse vibrations of a simply supported rectangular plate is solved to illustrate the method. Research was supported by the National Aeronautics and Space Administration under NASA Contract Nos. NASI-17070 and NASI-18107 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23665-5225. nl @3BdOS#i I° Introduction Rotating beam configurations have traditionally been used to study the vibrations and aeroelastic stability of rotating structures such as helicopter rotor blades and propeller blades. More recently, models involving elastic plates have been proposed to include the effects of spanwise variations in material properties. The fourth-order boundary value problems associated with both the beam and plate models do not, in general, have useful closed form solutions. Consequently, most theoretical work on these problems has been asymptotic or numerical in nature. In one approach to the numerical solution of these problems, harmonic time dependence is assumed to reduce the governing partial differential equation to a differential equation in space variables which includes an eigenvalue. For beam models, this is an ordinary differential equation. The fundamental derivative which represents beam curvature may now be taken as a new dependent variable, and the eigenvalue problem for the beam can be reformulated as an integro-differential equation (White & Malatino, 1975; Kvaternik, White, & Kaza, 1978; Lakin, 1982). This equation may be conveniently expressed using integral, differential, and boundary evaluation operators. The operator equation for the continuous solution may further be converted to a matrix operator equation for a finite-dimensional solution vector by evaluating the continuous equation at a finite set of discrete grid points which span the interval of interest. A key question is now the manner in which the matrix operators are approximated. For beam models, one method for approximating the integral and differential operators involves the use of integrating matrices (Vakhitov, 1966; Hunter, 1970; Lakin, 1979) and differentiating matrices (Hunter &