Meshfree Modeling and Analysis

Torsion of a Rod

We now apply RFM to the classical torsion problem for a rod with the cross section shown in Figure 1(a). This is a textbook problem [2] with many good approximations already known. For example, an approximate analytic expression for torque in terms of parameters r, b, and a has been derived for the same domain in [3] and will be used here for comparison. It is well known that the torsion problem may be reduced to the boundary value problem with Poisson equation and homogeneous Dirichlet boundary conditions. We will represent the solution u as a product of two functions: (1) function which takes on zero value on the boundary of the domain (Figure 1(b)) and (2) unknown function which may be approximated by linear combination of basis functions with unknown coefficients. Different sets of the coefficients correspond to different functions, but all of them satisfy homogeneous Dirichlet boundary condition exactly. For example, Figure 2 shows function u with randomly assigned coefficients. The set of the coefficients which corresponds to the solution of the torsion problem may be obtained via any variational or projection method. Figure 3(a,b) show the stress functions given by the Ritz's method on 40x40 grid of bicubic B-splines for different values of parameter b. Once we have a stress function, we can compute shear stresses (Figure 4) which are the partial derivatives of the stress function or compute the torque (Figure 5) - the integral of a stress function over the geometric domain. Figure 5 shows a good agreement of the torque determined by RFM and the torque obtained by a closed-form approximation. For more information see the reference [4]. (a) (b) Figure 1: Two dimensional domain and function vanishes on the boundary of the domain Figure 2: Function with randomly assigned coefficients (a) (b) Figure 3: Stress functions computed by RFM for b=0.2 (a) and b=0.5 (b) (a) (b) Figure 4: Shear stresses predicted by RFM Figure 5: Comparison of torque predicted by RFM with torque obtained by closed-form approximation

References

[1] V.L. Rvachev, T.I. Sheiko, V. Shapiro, I. Tsukanov, Transfinite Interpolation Over Implicitly Defined Sets, Computer Aided Geometric Design, Vol. 18, No. 4, 2001, pp.195-220 [2] Timoshenko, S. and Goodier, N.  Theory of Elasticity. McGraw-Hill Book Company, New York, 1970 [3] Pilkey, W. D.  Formulas for stress, strain, and structural matrices. John Wiley & Sons, 1994 [4] V.L.Rvachev, T.I.Sheiko, V.Shapiro and I.Tsukanov, On Completeness of RFM Solution Structures, Computational Mechanics, special issue on meshfree methods, Vol. 25, 2000, pp. 305-316