Award Date

5-15-2026

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Committee Member

Angel Muleshkov

Second Committee Member

Zhonghai Ding

Third Committee Member

Xin Li

Fourth Committee Member

Stephen Lepp

Number of Pages

156

Abstract

For decades, the evaluation of steady state seepage beneath asymmetrical cofferdams has relied on numerical methods, such as Finite Element Methods (FEM), Finite Difference Method (FDM), Finite Volume Method (FVM), Mesh Reduction Method (MRM), Meshless Method (MM), Boundary Element Method (BEM), or geometric idealizations, most notably Griffiths' vertical Method of Fragments assumption. While exact closed form solutions via Schwarz-Christoffel (SC) conformal mapping have been well established for symmetrical geometries (Banerjee and Muleshkov), the generalized asymmetrical case has historically remained an intractable mathematical frontier. The primary barrier to an exact analytical solution has been the "crowding problem," a numerical phenomenon where physical asymmetry forces preimage vertices in the auxiliary t−plane to cluster exponentially. This clustering triggers a catastrophic loss of significance and unresolvable division by zero singularities within standard 16-digit double precision computational solvers.

This dissertation presents the first exact, fully generalized analytical solution for groundwater flow beneath unsymmetrical cofferdams. By mapping the complex physical geometry (z−plane) through an intermediate half plane and into the complex potential domain (ω−plane), the boundary value problem is reduced to a highly nonlinear 5 x 5 system of Legendre Elliptic Integrals of the first, second, and third kinds. To overcome the historic crowding wall, this work introduces a novel topological constraint mapping (the u-space transformation) coupled with a three-stage, arbitrary-precision numerical pipeline operating at up to 50-digit precision. This computational architecture strictly enforces physical geometric ordering, shielding the solver from singularity collapses and allowing for the exact evaluation of deeply crowded roots.

Utilizing the newly resolved parameters, the exact total discharge (Q), critical exit gradients, and exact flow split kinematics are derived without reliance on finite element meshes or vertical streamline approximations. Finally, the exact analytical flow nets are generated using dynamically scaled numerical integration. The results are rigorously benchmarked against prior numerical and symmetrical SC models, establishing a new mathematical standard for evaluating extreme asymmetric seepage scenarios while proving the viability of complex conformal mapping in highly skewed geotechnical domains.

Keywords

Asymmetrical; Cofferdam; Conformal Mapping; Elliptic Integrals; Exact Solution; Water Seepage

Disciplines

Mathematics | Other Mathematics | Physical Sciences and Mathematics

File Format

PDF

File Size

2300 KB

Degree Grantor

University of Nevada, Las Vegas

Language

English

Rights

IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/

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