Meshfree Modeling and Analysis

Linear Elasticity

This example shows solution of a linear elasticity problem via the R-function meshfree method [1, 2, 4, 5]. Figure 1 shows the loaded cantilever beam which is fixed at two circular holes. This prevents motion of the boundary points of these holes in both horizontal and vertical directions. The displacement is represented by a vector RFM solution structure satisfying the kinematic and loading constraints. This vector solution structure transfinitely interpolates [6] the prescribed boundary conditions for each component of the displacement vector. The vector solution structure is automatically differentiated [7, 8] and then integrated [10] over the geometric domain in order to minimize the potential energy of the mechanical system. This solution procedure results in numerical values of the coefficients in the vector solution structure. Figure 2 presents the distribution of the components of the displacement vector and Figure 3 shows the distributions of the principal stresses in the beam. Figure 1: The loaded cantilever beam (a) (b) Figure 2: The distribution of (a) horizontal and (b) vertical components of the displacement vector (a) (b) Figure 3: The distribution of the principal stresses in the beam shown in Figure 1

References

[1] V. L. Rvachev. Theory of R-functions and Some Applications. Naukova Dumka, 1982. In Russian. [2] Theory of R-functions and applications: A primer. V. Shapiro, Technical Report, Cornell University. [3] V. Shapiro and I. Tsukanov, Implicit Functions with Guaranteed Differential Properties,In Proceedings of the Fifth ACM Symposium on Solid Modeling and Applications, June 1999, Ann Arbor, MI [4] V. Shapiro and I. Tsukanov, Meshfree Simulation of Deforming Domains, Computer-Aided Design , Vol. 31, No. 7, 1999, pp. 459-471 [5] 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 [6] 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 [7] I. Tsukanov and M. Hall, Data Structure and Algorithms for Fast Automatic Differentiation, Technical Report SAL 2002-5, Spatial Automation Laboratory, University of Wisconsin-Madison, http://sal-cnc.me.wisc.edu, July 2002. [8] I. Tsukanov, M.Hall, Fast Forward Automatic Differentiation Library (FADLib): A User Manual, Technical Report SAL 2000-4, Spatial Automation Laboratory, University of Wisconsin-Madison, http://sal-cnc.me.wisc.edu, December 2000. [9] I. Tsukanov, V. Shapiro, S. Zhang, A Meshfree Method for Incompressible Fluid Dynamics Problems, SAL Technical report 2002-1, University of Wisconsin-Madison, http://sal-cnc.me.wisc.edu, January 2002 [10] I. Tsukanov and V. Shapiro, The architecture of SAGE - a meshfree system based on RFM, Engineering with Computers, 2002. Accepted for publication