arXiv subjects

Luobin Lin

Publications and source records attributed to Luobin Lin.

4 recordsLinked to original sources

arXiv2024

Bidirectional conformal mapping for over-break and under-break tunnelling and its application in complex variable method

Over-break and under-break excavation is very common in practical tunnel engineering with asymmetrical cavity contour, while existing conformal mapping schemes of complex variable method generally focus on tunnelling with theoretical and symmetrical cavity contour. Besides, the solution strategies of existing conformal mapping schemes for noncircular tunnel generally apply optimization theory, and are thereby mathematically complicated. This paper proposes a new bidirectional conformal mapping for over-break and under-break tunnels of asymmetrical contours by incorporating Charge Simulation Method, which only involves a pair of forward and backward linear systems, and is therefore logically straight-forward, computationally efficient, and practically easy in coding. New numerical strategies are developed to deal with possible sharp corners of cavity by small arc simulation and densified collocation points. Several numerical examples are presented to illustrate the geometrical usage of the new bidirectional conformal mapping. Furthermore, the new bidirectional conformal mapping is embedded into two complex variable solutions of under-break shallow tunnelling in gravitational geomaterial with reasonable far-field displacement. The respective result comparisons with finite element solution and existing analytical solution show good agreements, indicating that the new bidirectional conformal mapping would extend the mechanical application range of the complex variable method in practical over-break and under-break tunnelling.

math.NA↗

arXiv2023

Reasonable mechanical model on shallow tunnel excavation to eliminate displacement singularity caused by unbalanced resultant

When considering initial stress field in geomaterial, nonzero resultant of shallow tunnel excavation exists, which produces logarithmic items in complex potentials, and would further lead to a unique displacement singularity at infinity to violate geo-engineering fact in real world. The mechanical and mathematical reasons of such a unique displacement singularity in the existing mechanical models are elaborated, and a new mechanical model is subsequently proposed to eliminate this singularity by constraining far-field ground surface displacement, and the original unbalanced resultant problem is converted into an equilibrium one with mixed boundary conditions. To solve stress and displacement in the new model, the analytic continuation is applied to transform the mixed boundary conditions into a homogenerous Riemann-Hilbert problem with extra constraints, which is then solved using an approximate and iterative method with good numerical stability. The Lanczos filtering is applied to the stress and displacement solution to reduce the Gibbs phenomena caused by abrupt change of the boundary conditions along ground surface. Several numerical cases are conducted to verify the proposed mechanical model and the results strongly validate that the proposed mechanical model successfully eliminates the displacement singularity caused by unbalanced resultant with good convergence and accuracy to obtain stress and displacement for shallow tunnel excavation. A parametric investigation is subsequently conducted to study the influence of tunnel depth, lateral coefficient, and free surface range on stress and displacement distribution in geomaterial.

math.NA↗

arXiv2023

Parallel generalized solutions of mixed boundary value problem on partially fixed unit annulus subjected to arbitrary traction

This paper provides two parallel solutions on the mixed boundary value problem of a unit annulus subjected to a partially fixed outer periphery and an arbitrary traction acting along the inner periphery using the complex variable method. The analytic continuation is applied to turn the mixed boundary value problem into a Riemann-Hilbert problem across the free segment along the outer periphery. Two parallel interpreting methods of the unused traction and displacement boundary condition along the outer periphery together with the traction boundary condition along the inner periphery respectively form two parallel complex linear constraint sets, which are then iteratively solved via a successive approximation method to reach the same stable stress and displacement solutions with the Lanczos filtering technique. Finally, four typical numerical cases coded by \texttt{FORTRAN} are carried out and compared to the same cases performed on \texttt{ABAQUS}. The results indicate that these two parallel solutions are both accurate, stable, robust, and fast, and validate that these two parallel solutions are numerically equivalent.

math.NA↗

arXiv2022

On solution of conformal mapping for a lower half plane containing a symmetrical noncircular cavity

In this paper, we provide a candidate solution to obtain the coefficients of the conformal mapping for a lower half plane containing a symmetrical noncircular cavity using penalty function method and modified Particle Swarm Method. The nonconvexity of the penalty function is proven via the concept of convex function and proof of contradiction. The solution procedure is presented very detailedly in pseudocodes to ensure that the solution can be fully repeated and further improved. The solution accuracy and efficiency are also discussed.

math.OC↗