Nonlinear boundary element method and its applications: a review

The Boundary Element Method (BEM) has long been recognized as an efficient numerical technique for the solution of linear boundary value problems, particularly those involving unbounded domains, free-surface effects and radiation conditions. However, its application to nonlinear problems poses additional challenges, including the loss of strict boundary-only discretization and the need for domain integration or numerical approximation strategies. As a result, nonlinear extensions of BEM have evolved more slowly than their linear counterparts, and related research remains comparatively fragmented across scientific domains. This paper presents a comprehensive review of nonlinear BEM, synthesizing the mathematical foundations, historical developments and application-specific methodologies reported in the literature. The review is organized around a structured thematic framework spanning solid mechanics, heat transfer, fluid mechanics, acoustics, electromagnetics and emerging multi-physics applications. Bibliometric analysis provides a supporting perspective on publication trends and research evolution. A comparative assessment highlights methodological strategies—such as domain–boundary element formulations, dual reciprocity, local domain methods and hybrid FEM–BEM couplings—and evaluates their applicability across problem classes. The review also identifies research gaps, including the scarcity of benchmark problems, limited software availability and a lack of systematic comparative studies across numerical frameworks. Overall, the survey demonstrates that while nonlinear BEM remains specialized relative to nonlinear FEM, it has matured into a viable and efficient alternative for problems involving unbounded domains, boundary-dominant phenomena and multi-field coupling. The findings of this review aim to support researchers, computational engineers and method developers by clarifying the current status, trends and future opportunities in nonlinear boundary element analysis.