Convex optimization problems inspired by geotechnical stability analysis

TL;DR

The paper addresses geotechnical stability analysis by reframing limit load, limit analysis, and shear strength reduction within a unified abstract convex optimization framework in $\mathbb{R}^n$, built around a convex potential $\mathcal{I}$ with gradient $F$ and linear growth. It develops continuation techniques and establishes deep links between the limit load FoS $t^*$, the LA problem with $\mathcal{I}_\infty$ and $t_\infty$, and SSR parametrizations $F_\lambda$ linked to a safety factor $\lambda^*$, including an indirect continuation approach and an LA-inspired solvability perspective. The framework yields rigorous solvability criteria, an advanced continuation methodology (including an indirect, $\omega$-controlled path for SSR), and a unified LA-SSR relationship, complemented by analytical and numerical examples including a 3D slope using Mohr-Coulomb plasticity. The practical impact lies in providing an accessible, mathematically rigorous toolkit for stability analysis that can be extended beyond geotechnics and used to study solvability and continuation in convex optimization scenarios. Overall, the work bridges engineering practice and convex analysis, enabling systematic estimation of FoS and synthesis of continuation strategies with potential applicability to broader variational problems with linear growth functionals.

Abstract

This paper is motivated by the limit load, limit analysis and shear strength reduction methods, which are commonly employed in geotechnical stability analysis or similar applications. The aim is to make these methods more approachable by introducing a unified framework based on abstract convex optimization and its parametric studies. We establish suitable assumptions on the abstract problems that capture the selected features of these methods and facilitate rigorous theoretical investigation. Further, we propose continuation techniques tailored to the resulting parametric problem formulations and show that the developed abstract framework could also be useful outside the domain of geotechnical stability analysis. The main results are illustrated with analytical and numerical examples. The numerical example deals with a 3D slope stability problem.

Figures (7)

• Figure 1: Scheme of the indirect continuation method with the control parameter $\omega$.
• Figure 1: Visualization of the function $\ell$, which relates $\lambda^*$ with LA.
• Figure 1: Geometry of the computational domain and the investigated mesh.
• Figure 2: Visualization of the sets $M_\lambda$ and $\mathcal{C}_\lambda$ for $\lambda_1<\lambda_2$.
• Figure 2: Continuation curves for the LL method (left) and the SSR method (right).

Theorems & Definitions (29)

• Lemma 1
• Proof 1
• Lemma 2
• Proof 2
• Theorem 1
• Proof 3
• Lemma 2
• Proof 4
• Theorem 3
• Proof 5