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Fig. 10. An infinite plate with a circular hole under a uniaxial load P. increase in the number of nodal points by an order of 0.7. These results clearly show that the proposed adaptive solution strategy is stable and efficient for accurate solution of elasticity problems.

Figure 10 An infinite plate with a circular hole under a uniaxial load P. increase in the number of nodal points by an order of 0.7. These results clearly show that the proposed adaptive solution strategy is stable and efficient for accurate solution of elasticity problems.

Related Figures (14)

The proposed adaptive refinement procedure based on error estimation is schematically shown in Fig. 2.
The proposed adaptive refinement procedure based on error estimation is schematically shown in Fig. 2.
Fig. 2. Proposed adaptive node enrichment procedure based on error estimation.
Fig. 2. Proposed adaptive node enrichment procedure based on error estimation.
Fig. 5. Initial, second and last adapted nodal configurations for the cantilever beam. Fig. 4. Voronoi diagram for initial nodal configuration.
Fig. 5. Initial, second and last adapted nodal configurations for the cantilever beam. Fig. 4. Voronoi diagram for initial nodal configuration.
The adaptive analysis starts with a regular distribution of 125 nodes as the initial configuration. The problem is solved on the initial configuration using the DLSM method, the errors are estimated and the nodal configuration adaptively refined using the Voronoi diagram of the initial configuration shown in Fig. 4. The process of simulating the problem on the resulting adapted nodal configuration, error estimation and construction of the adapted refined configuration is repeated until the global value of the error estimator is nearly constant. The initial nodal distributions along with the third and last adapted nodal config- urations are shown in Fig. 5. Starting with 125 regularly dis- tributed nodes, the computation ends at fifth adaptive step with the configuration containing 592 nodes. Fig. 6 shows the vertical
The adaptive analysis starts with a regular distribution of 125 nodes as the initial configuration. The problem is solved on the initial configuration using the DLSM method, the errors are estimated and the nodal configuration adaptively refined using the Voronoi diagram of the initial configuration shown in Fig. 4. The process of simulating the problem on the resulting adapted nodal configuration, error estimation and construction of the adapted refined configuration is repeated until the global value of the error estimator is nearly constant. The initial nodal distributions along with the third and last adapted nodal config- urations are shown in Fig. 5. Starting with 125 regularly dis- tributed nodes, the computation ends at fifth adaptive step with the configuration containing 592 nodes. Fig. 6 shows the vertical
Fig. 7. Stresses ox, along upper surface of the cantilever beam. Fig. 6. Vertical displacement along the central line of the cantilever beam.
Fig. 7. Stresses ox, along upper surface of the cantilever beam. Fig. 6. Vertical displacement along the central line of the cantilever beam.
Fig. 9. Convergence curve of stresses in the cantilever beam under the end load. Table 1 Log (error norm) of the solutions for different of nodal distribution (cantilever beam). displacement along the central line of the beam for three nodal configurations shown. Stress ox, along the upper surface of the cantilever beam is plotted in Fig. 7 for three nodal configurations where it is shown that a very good agreement with the analytical solution is obtained. The contours of o,, stress for the last configuration with 592 nodes are compared with analytical solution in Fig. 8. The convergence rate is also shown in Fig. 9. Table 1 shows the displacement and the stress error norms calculated for the initial and adapted nodal configurations. It is seen that the proposed adaptive refinement process have been able to reduce the error norms of the displacement and the stress by 3.5 and 2 order of magnitude, respectively, at the expense of an
Fig. 9. Convergence curve of stresses in the cantilever beam under the end load. Table 1 Log (error norm) of the solutions for different of nodal distribution (cantilever beam). displacement along the central line of the beam for three nodal configurations shown. Stress ox, along the upper surface of the cantilever beam is plotted in Fig. 7 for three nodal configurations where it is shown that a very good agreement with the analytical solution is obtained. The contours of o,, stress for the last configuration with 592 nodes are compared with analytical solution in Fig. 8. The convergence rate is also shown in Fig. 9. Table 1 shows the displacement and the stress error norms calculated for the initial and adapted nodal configurations. It is seen that the proposed adaptive refinement process have been able to reduce the error norms of the displacement and the stress by 3.5 and 2 order of magnitude, respectively, at the expense of an
Fig. 12. Three nodal configurations (second example). (a) Initial nodal configuration (141 nodes), (b) second adapted nodal configuration (345 nodes) and (c) the las adapted nodal configuration (857 nodes). Fig. 13. Horizontal displacement along circular hole (r=a). Fig. 11. Voronoi diagram for initial nodal configuration (second example).
Fig. 12. Three nodal configurations (second example). (a) Initial nodal configuration (141 nodes), (b) second adapted nodal configuration (345 nodes) and (c) the las adapted nodal configuration (857 nodes). Fig. 13. Horizontal displacement along circular hole (r=a). Fig. 11. Voronoi diagram for initial nodal configuration (second example).
Fig. 14. Normal stress o,, along x=0 (second example). The process begins with the simulation of the problem on an irregular distribution of 141 nodes as the initial nodal configura- tion. Starting with 141 irregularly distributed nodes, the compu- tation ends at fourth adaptive step with an adapted distribution of 857 nodes. The Voronoi diagram of initial nodal configuration is shown in Fig. 11. Fig. 12 shows the initial nodal configurations along with the third and last adapted configurations. The hor- izontal displacement along circular hole for three nodal config- urations is plotted in Fig. 13. Stress o,, along left boundary of the domain is shown in Fig. 14 for three nodal configurations showing here G is the shear modulus, « = (3—v)/(1 + v) with v representing the Poisson’s ratio. Due to symmetry, only the upper right square quadrant of the plate is modeled. The edge length of the square is 5a, with a being the radius of the circular hole. The Dirichlet boundary condition is imposed on the left and bottom boundaries and the tractions are applied to the top and right edges. The problem is solved under a plane stress condition with the following constants: P=1, E=1000 and v=0.3.
Fig. 14. Normal stress o,, along x=0 (second example). The process begins with the simulation of the problem on an irregular distribution of 141 nodes as the initial nodal configura- tion. Starting with 141 irregularly distributed nodes, the compu- tation ends at fourth adaptive step with an adapted distribution of 857 nodes. The Voronoi diagram of initial nodal configuration is shown in Fig. 11. Fig. 12 shows the initial nodal configurations along with the third and last adapted configurations. The hor- izontal displacement along circular hole for three nodal config- urations is plotted in Fig. 13. Stress o,, along left boundary of the domain is shown in Fig. 14 for three nodal configurations showing here G is the shear modulus, « = (3—v)/(1 + v) with v representing the Poisson’s ratio. Due to symmetry, only the upper right square quadrant of the plate is modeled. The edge length of the square is 5a, with a being the radius of the circular hole. The Dirichlet boundary condition is imposed on the left and bottom boundaries and the tractions are applied to the top and right edges. The problem is solved under a plane stress condition with the following constants: P=1, E=1000 and v=0.3.
Fig. 15. Normal stress o,, (second example). (a) Exact solution and (b) numerical solution at final adaptive step (857 nodes) Log (Error norm) of the solutions for different of nodal distribution (infinite plate). Fig. 16. Convergence rate of stresses for the infinite plate with circular hole.
Fig. 15. Normal stress o,, (second example). (a) Exact solution and (b) numerical solution at final adaptive step (857 nodes) Log (Error norm) of the solutions for different of nodal distribution (infinite plate). Fig. 16. Convergence rate of stresses for the infinite plate with circular hole.
an excellent agreement with the analytical solution. The contours of 6x, stress for the final configuration of 857 nodes are compared with the analytical solution in Fig. 15. The convergence rate of the proposed method is also shown in Fig. 16 indicating stability and convergent properties of the proposed method. Table 2 shows the
an excellent agreement with the analytical solution. The contours of 6x, stress for the final configuration of 857 nodes are compared with the analytical solution in Fig. 15. The convergence rate of the proposed method is also shown in Fig. 16 indicating stability and convergent properties of the proposed method. Table 2 shows the
Related topics:
Mechanical EngineeringCivil EngineeringMathematicsApplied MathematicsNumerical SimulationsComputational MechanicsMeshfree MethodsElasticity (Solid Mechanics)Numerical AnalysisRefinement (Formal Methods)
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