Global–Local Non Intrusive Analysis with 1D to 3D Coupling: Application to Crack Propagation and Extension to Commercial Software
Abstract
:1. Introduction
2. Methodology
2.1. Primal–Dual Global–Local Analysis
- First, the global problem is solved by obtaining the displacements :
- Second, the auxiliary problem is solved by imposing the displacements and solve for the reaction forces in the interface zone:The can be obtained directly in some software, allowing the obtainment of the reaction forces from embedded structures within a global problem.The displacements can be extracted from the solution of the global problem using the following relation:
- Third, the local problem is solved by imposing the displacements on the interface of the local modelHence, the reaction forces of the local model in the interface, solved by means of a nonlinear solver such as Arc Length Method [44] or Newthon–Raphson Method, is obtained from the following equation:
- Fourth, the correction forces that will be applied to the global model are calculated:
- Fifth, the residual force is calculated and the error of the solution obtained in the iteration is estimated:
- Finally, a relaxation scheme is considered, obtaining the following correction force:This relaxation allows for better convergence, for example, when Aitken relaxation method is used [19].
2.2. Case Study
- Local Mesh 1 (L.M. 1): Length 500 mm
- Local Mesh 2 (L.M. 2): Length 750 mm
- Local Mesh 3 (L.M. 3): Length 1000 mm
- The direction of propagation is taken into account, with a tangent vector (0,0,1) and normal vector (1,0,0) with the function of Code_Aster.
- The propagation is calculated internally, calculating the energy release rate using the intensity factors with the function of Code_Aster for a predefined number of propagation steps (function ).
2.3. Projection of Displacements from the Global 1D Model to the Local 3D Model
- 1.
- To calculate the displacement generated from the rotation resulting from bending , kinematic compatibility is considered, using a non-deformable finite element (solid face with no warping) and rotating with respect to the centroid. The face of the 3D element analyzed has a maximum distance from the centroid and when rotated it is maintained, producing a displacement , as shown in Figure 8.
- 2.
- 3.
- Finally, the values of and are found, which would be the effects that must be considered due the bending rotations, leading to the following expressions:
- 1.
- 2.
- Then, the values of and are found, which would be the effects on the displacement due to torsional moment, leading to:
3. Implementation of the Methodology in Code_Aster
3.1. Validation of the Implementation in Code_Aster
3.2. Effect of the Local Model Size
- Stagnation of the solution: If the iterative analysis presents a divergent error (increasing with each iteration), jumps between an error greater than the tolerance, or does not converge within a maximum number of iterations, i.e., 50 iterations in the present study.
- Failed crack propagation analysis (XFEM internal procedure): Crack propagation in Code_Aster is calculated using the rate of energy release (G) method, using the built-in function . This method calculates the intensity stress factors (K) evaluating the bilinear form of G with the asymptotic solution of Westergaard. In addition, an error indicator is obtained by comparing the difference between G and Irwin’s energy release rate (), as shown in Equation (32) [46].If the error calculated using the Equation (32) is greater than 50%, the analysis stops and displays an alert message as presented in [46]. This is the case for the L.M.1 local mesh with the 75 mm initial crack length, affecting the convergence of XFEM method and, therefore, the overall convergence of the global–local analysis. More information with respect to the convergence of the XFEM crack propagation method can be found in [47,48].
4. Implementation with a Commercial Software
4.1. Validation of the Implementation in SAP 2000
4.2. Methodology Extension to 3-Story Building
- Total height: H = 300 mm.
- Flange width: B = 200 mm.
- Flange thickness: tf = 10 mm.
- Web thickness: tw = 6 mm.
- Material: Grade 50 quality steel.
5. Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Local Mesh Model | Initial Crack Length in the Local Model | ||||
---|---|---|---|---|---|
Linear | 25 mm | 50 mm | 75 mm | ||
Local Mesh 1 | % disp. error | 6.35% | 5.96% | 5.91% | non conv. |
Local Mesh 2 | % disp. error | 6.44% | 5.93% | 5.91% | 6.62% |
Local Mesh 3 | % disp. error | 6.39% | 5.95% | 5.98% | 6.44% |
Local Mesh Model | Initial Crack Length in the Local Model | ||||
---|---|---|---|---|---|
Linear | 25 mm | 50 mm | 75 mm | ||
Local Mesh 1 | % error disp. | 8.94% | 8.41% | 8.32% | non conv. |
Local Mesh 2 | % error disp. | 8.87% | 8.33% | 8.32% | 9.11% |
Local Mesh 3 | % error disp. | 8.79% | 8.32% | 8.37% | non conv. |
Model | Crack Tip Displ. (mm) | Execution Time (s) |
---|---|---|
3D Monolithic | 6.01 | 2231 |
GL w/SAP 2000 | 5.89 | 391 |
Diff. r/Monolithic | 1.6% | 82% |
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Jaque-Zurita, M.; Hinojosa, J.; Fuenzalida-Henríquez, I. Global–Local Non Intrusive Analysis with 1D to 3D Coupling: Application to Crack Propagation and Extension to Commercial Software. Mathematics 2023, 11, 2540. https://doi.org/10.3390/math11112540
Jaque-Zurita M, Hinojosa J, Fuenzalida-Henríquez I. Global–Local Non Intrusive Analysis with 1D to 3D Coupling: Application to Crack Propagation and Extension to Commercial Software. Mathematics. 2023; 11(11):2540. https://doi.org/10.3390/math11112540
Chicago/Turabian StyleJaque-Zurita, Matías, Jorge Hinojosa, and Ignacio Fuenzalida-Henríquez. 2023. "Global–Local Non Intrusive Analysis with 1D to 3D Coupling: Application to Crack Propagation and Extension to Commercial Software" Mathematics 11, no. 11: 2540. https://doi.org/10.3390/math11112540