Highly accurate solutions of Motz's and the cracked beam problems

Abstract

For Motz's problem and the cracked beam problem, the collocation Trefftz method is used to seek their approximate solutions uN=∑i=0NDiri+(1/2)cos(i+(1/2))θ, where Di are the expansion coefficients. The high-order Gaussian rules and the central rule are used in the algorithms, to link the collocation method and the least squares method, and to provide exponential convergence rates of the obtained solutions. Compared with the solutions in the previous literature, our Motz's solutions are more accurate and the leading coefficient D0 using the Gaussian rule with six nodes arrives at 17 significant (decimal) digits. Similarly for the cracked beam problem, the collocation Trefftz method also provides the highly accurate solutions, and D0 with 17 significant digits by the Gaussian rules. This papers proves that when the rules of quadrature involved have the relative errors less than three quarters, the solution form the collocation Trefftz method may converge exponentially. Such an analysis supports the collocation Trefftz method to become theoretically the most accurate method for Motz's and the cracked beam problems.