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ICOME2006. Torsional rigidity of a circular bar with multiple circular inclusions using a null-field integral approach. Ying-Te Lee, Jeng-Tzong Chen and An-Chien Wu. Date: November 14-16. Place: Hefei, China. Outlines. 1. Introduction. 2. Problem statement. 3. Method of solution. 4.

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## ICOME2006

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**ICOME2006**Torsional rigidity of a circular bar withmultiple circular inclusions using a null-field integral approach Ying-Te Lee, Jeng-Tzong Chen and An-Chien Wu Date: November 14-16 Place: Hefei, China**Outlines**1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Motivation Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Meshless method BEM / BIEM Treatment of singularity and hypersingularity Boundary-layer effect Convergence rate Ill-posed model**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Motivation BEM / BIEM Improper integral Singularity & hypersingularity Regularity Fictitious BEM Bump contour Limit process Fictitious boundary Achenbach et al. (1988) Null-field approach Guiggiani (1995) Gray and Manne (1993) Collocation point CPV and HPV Ill-posed Waterman (1965)**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Present approach Degenerate kernel Fundamental solution No principal value CPV and HPV • Advantages of present approach • No principal value • Well-posed model • Exponential convergence • Free of mesh**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Literature review Analytical solutions for problems with circular boundaries Those analytical methods are only limited to doubly connected regions.**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Literature review Fourier series approximation However, no one employed the null-field approach and degenerate kernel to fully capture the circular boundary.**y**B0 B2 B1 a2 a1 a0 x B3 Bi ai a3 a4 B4 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Problem statement A circular bar with circular inclusions**B0**B0 Satisfy B2 B2 B1 B1 B3 B3 Bi Bi B4 B4 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Domain superposition A circular bar with circular holes Each circular inclusion problem**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Boundary integral equation and null-field integral equation Interior case Exterior case Degenerate (separate) form**cosnθ, sinnθ**boundary distributions kth circular boundary 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Degenerate kernel and Fourier series x Expand fundamental solution by using degenerate kernel s O x Expand boundary densities by using Fourier series**collocation point**1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Adaptive observer system r2,f2 r0 ,f0 r1 ,f1 rk,fk**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Comparisons of conventional BEM and present method**ex**R0 R1 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Case 1: A circular bar with an eccentric inclusion Ratio: Torsional rigidity: GT : total torsion rigidity GM : torsion rigidity of matrix GI : torsion rigidity of inclusion**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Results of case 1 Torsional rigidity versus number of Fourier series terms Torsional rigidity versus shear modulus of inclusion**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Results of case 1 Torsional rigidity of a circular bar with an eccentric inclusion**ex=0.5**R0=1 R1=0.3 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Case 2: limiting caseA circular bar with one circular hole**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Results of case 2 Torsional rigidity of a circular bar with an eccentric hole**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Stress calculation tm t D: External diameter of the tube tm: The maxium wall thickness (eccentricity)**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Stress calculationalong outer and inner boundary at boundaries for λ=0.3 and p=0.4 (0.0%) (0.3%) (0.1%) (0.0%) (0.0%) (0.0%) (0.0%) (1.5%) (0.4%) (0.6%)**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Stress calculationfor point in the center line alnog lines and for λ=0.3 and p=0.4 (0.0%) (0.0%) (0.1%) (0.2%) (0.1%) (0.5%) (0.1%) (0.5%) (0.1%) (0.0%) (0.3%) (0.6%)**1. Introduction**2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Concluding remarks 1. A systematic approach was proposed for torsion problems with circular inclusionsby usingnull-field integral equation in conjunction with degenerate kernel and Fourier series. 2. A general-purpose program for multiple circular inclusions of various radii, numbers and arbitrary positions was developed. 3. Onlya few number of Fouries series terms for our exampleswere needed on each boundary, and for more accurate results of torsional rigidity with error less than 2 %. 4. Fourgains of our approach, (1) free of calculating principal value, (2) exponential convergence, (3) free of meshand (4) well-posed model**The End**Thanks for your kind attention Welcome to visit the web site of MSVLAB http://ind.ntou.edu.tw/~msvlab**: the shear modulus**: angle of twist per unit length 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Torsion problem Following the theory of Saint-Venant torsion, we assume Displacement fields: Strain components: Stress components: Equilibrium equation:

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