Comparative Analysis of ALE Method Implementation in Time Integration Schemes for Pile Penetration Modeling
Abstract
:1. Introduction
2. Contact Kinematics
2.1. Normal Conditions
- No penetration occurs
- Compressive contact stress
- The normal contact stress exists only when the surfaces are in contact () and when the surfaces are separated (). These complementary conditions are formulated as .
2.2. Contact with Friction
- Contact with adherence
- Contact with slip
3. Numerical Methods
3.1. Numerical Time Integration Schemes
3.1.1. Explicit Time Integration
3.1.2. Implicit Time Integration
3.2. Arbitrary Lagrangian–Eulerian Method
4. Validation of the Numerical Model
4.1. Field Experiments
4.2. Finite Element Model
- Fine mesh: Approximately half of the pile shaft radius.
- Medium mesh: Equal to the pile shaft radius.
- Coarse mesh: Two times the pile shaft radius.
4.3. Results and Discussion
4.3.1. Mesh Size Effect
4.3.2. Pile-End Bearing Capacity
4.3.3. Radial Total Stress During Installation
5. ALE Method in Explicit and Implicit Time Integration
5.1. Numerical Model
5.2. Results and Discussion
5.2.1. Deformation of the Mesh at Different Depths of Installation
5.2.2. Effect of Dilation Angle on Pile Resistance
5.2.3. Effect of Pile Dimensions on Pile Resistance
5.2.4. Influence of Soil–Pile Interface Friction on Pile Resistance
5.2.5. Computational Efficiency
6. Conclusions
- The ALE method effectively addressed mesh distortion challenges in both the explicit and implicit schemes without significant computational overhead.
- The computational results from both explicit and implicit schemes indicate that higher interface friction coefficients consistently yield greater total resistance values. Furthermore, the findings indicate that an increase in pile dimension is associated with an enhancement in its overall resistance, primarily due to optimization of the pile-end and shaft friction resistances.
- The explicit time integration scheme demonstrated superior stability and robustness, particularly in cases involving higher friction coefficients, despite characteristic oscillations in the solutions. While the implicit scheme produced smoother solutions, it showed limitations at higher friction coefficients, suggesting reduced effectiveness under complex soil–pile interaction conditions.
- A higher dilation angle () enhances pile resistance by increasing the confining and shaft stress, whereas a lower dilation angle () leads to reduced confinement and lower resistance. These findings align with fundamental principles of soil mechanics, where dilation contributes to increased shear strength and resistance due to volumetric constraints, thereby enhancing the pile resistance.
- The explicit method exhibited lower CPU time requirements, compared to the implicit approach.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Mesh | Element Sizes (m) | Total N Elements |
---|---|---|
Coarse | 0.1 | 425 |
Medium | 0.05 | 1249 |
Fine | 0.025 | 3862 |
Property | Value |
---|---|
Weathered Firm Crust | |
Elastic modulus (MPa) | 3 |
Poisson’s ratio, | 0.2 |
Cohesion (kPa) | 6 |
Friction angle, | 30° |
Dilation angle, | 0° |
Dry unit weight (kg/m3), | 1900 |
Bothkennar Clay | |
0.3 | |
Poisson’s ratio, | 0.2 |
0.02 | |
M | 1.5 |
Initial void ratio, | 2 |
Permeability (m/s), k | |
OCR | 1.5 |
Lateral stress coefficient, | 0.5 |
Saturated unit weight (kg/m3), | 1650 |
Property | Value |
---|---|
Pile | |
Elastic modulus (GPa) | 25 |
Poisson’s ratio, | 0.3 |
Soil | |
Elastic modulus (MPa) | 10 |
Poisson’s ratio, | 0.3 |
Cohesion (kPa) | 1 |
Friction angle, | 37° |
Dilation angle, | 0–8° |
Density (kg/m3), | 1700 |
Schemes | CPU Time (s) | Increments |
---|---|---|
Explicit | 370 | 106,051 |
Implicit | 1569 | 10,014 |
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Bourokba, I.B.; Berga, A.; Staubach, P.; Terfaya, N. Comparative Analysis of ALE Method Implementation in Time Integration Schemes for Pile Penetration Modeling. Math. Comput. Appl. 2025, 30, 58. https://doi.org/10.3390/mca30030058
Bourokba IB, Berga A, Staubach P, Terfaya N. Comparative Analysis of ALE Method Implementation in Time Integration Schemes for Pile Penetration Modeling. Mathematical and Computational Applications. 2025; 30(3):58. https://doi.org/10.3390/mca30030058
Chicago/Turabian StyleBourokba, Ihab Bendida, Abdelmadjid Berga, Patrick Staubach, and Nazihe Terfaya. 2025. "Comparative Analysis of ALE Method Implementation in Time Integration Schemes for Pile Penetration Modeling" Mathematical and Computational Applications 30, no. 3: 58. https://doi.org/10.3390/mca30030058
APA StyleBourokba, I. B., Berga, A., Staubach, P., & Terfaya, N. (2025). Comparative Analysis of ALE Method Implementation in Time Integration Schemes for Pile Penetration Modeling. Mathematical and Computational Applications, 30(3), 58. https://doi.org/10.3390/mca30030058