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Research Papers

# Hybrid Method of Engineering Analysis: Combining Meshfree Method with Distance Fields and Collocation Technique

[+] Author and Article Information
Igor Tsukanov1

Sudhir R. Posireddy

Department of Mechanical and Materials Engineering,  Florida International University, 10555 West Flagler Street, Miami, FL 33174sposi001@fiu.edu

1

Corresponding author.

J. Comput. Inf. Sci. Eng 11(3), 031001 (Jul 22, 2011) (9 pages) doi:10.1115/1.3572035 History: Received May 22, 2010; Revised October 11, 2010; Published July 22, 2011; Online July 22, 2011

## Abstract

This paper describes a numerical technique for solving engineering analysis problems that combine radial basis functions and collocation technique with meshfree method with distance fields, also known as solution structure method. The proposed hybrid technique enables exact treatment of all prescribed boundary conditions at every point on the geometric boundary and can be efficiently implemented for both structured and unstructured grids of basis functions. Ability to use unstructured grids empowers the meshfree method with distance fields with higher level of geometric flexibility. By providing exact treatment of the boundary conditions, the new approach makes it possible to exclude boundary conditions from the collocation equations. This reduces the size of the algebraic system, which results in faster solutions. At the same time, the boundary collocation points can be used to enforce the governing equation of the problem, which enhances the solution’s accuracy. Application of the proposed method to solution of heat transfer problems is illustrated on a number of benchmark problems. Modeling results are compared with those obtained by the traditional collocation technique and meshfree method with distance fields.

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## Figures Figure 1

Geometric domain Figure 2

(a) Distance field to the outer circle of the geometric domain shown in Fig. 1. (b) Distance field to the inner circle of the geometric domain shown in Fig. 1. (c) Distance field to the boundary of a geometric domain in Fig. 1 is constructed combining the distance fields in (a) and (b) using R-functions. (d) Function interpolating the boundary conditions (Eq. 19). Figure 3

Dirichlet boundary value problem: effect of varying shape parameter c on solution error for different grid sizes: (a) 20×19 size grid and (b) 40×39 size grid Figure 4

Distribution of the solution error obtained by (a) traditional collocation approach, (b) hybrid approach with no collocation points on the boundary, and (c) hybrid approach using boundary collocation points to enforce the differential equation. Numerical simulations were performed using c=0.2. Figure 5

Dirichlet boundary value problem: dependence of solution error on the number of radial basis functions Figure 6

Robin boundary value problem: effect of varying shape parameter c on solution error for different grid sizes for convective boundary conditions: (a) 20×19 size grid and (b) 40×39 size grid Figure 7

Robin boundary value problem: dependence of the solution error on the number of radial basis functions Figure 8

Geometric domain for modeling a temperature field Figure 9

Temperature distributions obtained by (a) traditional collocation technique, (b) proposed hybrid meshfree technique, (c) meshfree method with distance fields, and (d) finite element analysis in Ansys 12.1 Figure 10

(a) Geometric domain; temperature distributions obtained by (b) traditional collocation technique, (c) proposed hybrid meshfree technique, (d) meshfree method with distance fields, and (e) finite element analysis in Ansys 12.1

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