Meshfree Modeling and Analysis

Thermo-elasticity

We illustrate the modeling of a multi-physics phenomena via the R-function meshfree method using thermo-elasticity problem as an example. An infinite steel rod is fixed at four circular holes as shown in Figure 1. Initially the rod is at the uniform temperature of 273K. Then the steady heat flux q is applied at the boundary of the central hole. The outer surface of the rod is under convective heat exchange with the environment whose temperature is 273K. The temperature and the components of the displacement vector are represented by the RFM solution structures satisfying the prescribed boundary conditions exactly [1, 4, 5, 6]. Each of these solution structures contains about 5000 degrees of freedom (bicubic B-splines defined over an uniform 72 x 72 rectangular grid). In order to solve the transient heat transfer problem the differential equation has been discretized by time using an implicit finite difference scheme [4]. At each time instance the quasi-steady heat transfer problem has been solved and the computed temperature field was used to model the distribution of the thermal stresses. Figure 2 shows the distribution of the temperature field and the principal stress s1 at several time instances. To see the animation click on the images. Figure 1: The cross section of an infinite rod (a) (b) (c) (d) (e) (f) Figure 2: The distribution of (a, c, e) temperature field and (b, d, f) pricipal stress s1 inside the rod at different time instances

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