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Key research themes
1. How can global geometric formulations improve the analysis of Lagrangian and Hamiltonian mechanics on curved manifolds such as two-spheres?
This research area focuses on developing intrinsic, coordinate-free formulations of Lagrangian and Hamiltonian dynamics on curved manifolds—in particular the product of multiple two-spheres—thus avoiding the local coordinate singularities and complicated trigonometric expressions inherent in traditional angle-based parameterizations. Such global geometric formulations provide elegant, compact equations of motion directly incorporating manifold geometry, with significant advantages for analysis and numerical computation in mechanical systems evolving on spherical geometries.
Key finding: Developed four types of Euler-Lagrange and Hamilton's equations expressed in a coordinate-free manner on the product of an arbitrary number of two-spheres; this intrinsic formulation avoids local coordinate singularities... Read more
2. What are the theoretical and computational benefits of employing augmented Lagrangian methods with second-order optimality guarantees in constrained nonlinear optimization?
This theme encompasses advances in augmented Lagrangian (AL) optimization methods that rigorously establish convergence to points satisfying second-order stationarity conditions under weak constraint qualifications and general problem settings beyond standard nonlinear programming. By incorporating second-order information in subproblem solvers and relaxing strict constraint qualification assumptions, these methods improve the quality of stationary points found, increase convergence robustness, and provide theoretical guarantees even with inexact subproblem solutions. This has implications across nonlinear, nonconvex, and constrained optimization problems encountered in machine learning, matrix optimization, and bi-level programming.
Augmented Lagrangians with constrained subproblems and convergence to second-order stationary points
Key finding: Presented a class of augmented Lagrangian algorithms allowing arbitrary partitioning of constraints into lower-level constraints enforced strictly within subproblems and upper-level constraints handled via penalties; methods... Read more
Key finding: Proved equivalence between primal quadratic growth condition of the classical augmented Lagrangian around local minimizers and the second-order sufficient optimality conditions (SOSC), and established analogous equivalence... Read more
Key finding: Extended augmented Lagrangian methods to a generalized nonlinear programming framework incorporating nonsmoothness and composite constraints; demonstrated derivation from the proximal point algorithm via local duality... Read more
3. How can alternative or deformed Lagrangians and inverse problem approaches deepen understanding and enable novel dynamics formulations for mechanical and physical systems?
This theme investigates approaches for generating alternative or deformed Lagrangian functions associated with given equations of motion, including time-dependent (non-autonomous) or nonlinear systems, and explores mathematical conditions under which such Lagrangians exist. These approaches leverage geometric methods, inverse problems of the calculus of variations, and deformation functions acting on a base Lagrangian to obtain dynamically equivalent or related systems. Insights facilitate the construction of constants of motion, understanding of nonlinear integrable systems, and provide new perspectives on dissipative or non-standard mechanical systems in physics and engineering.
Key finding: Extended results on deformation of autonomous Lagrangian functions by scalar functions φ(L) to the non-autonomous case, characterized by time as a coordinate rather than a curve parameter; derived intrinsic geometric... Read more
Key finding: Further developed the non-autonomous geometric framework for Lagrangian deformation, analyzing the roles of jets and one-dimensional distributions replacing vector fields in time-dependent systems; established conditions for... Read more
Key finding: Utilized the Helmholtz inverse problem conditions to investigate existence of alternative, non-gauge equivalent Lagrangians for integrable nonlinear systems; established connections between constants of motion and alternative... Read more
Key finding: Demonstrated a derivation of Euler’s rigid body rotation equations starting from classical Lagrangian dynamics parametrized by generalized coordinates such as 3-2-1 and 3-1-3 Euler angles and Euler parameters (quaternions);... Read more
Key finding: Formulated integrability conditions for implicit differential systems defined by Lagrangian submanifolds in the tangent bundle of a symplectic manifold, presenting necessary and sufficient conditions for complete... Read more
4. How can Lagrangian approaches be adapted to numerical simulation of complex fluid dynamics involving non-Newtonian and turbulent behaviors?
This theme covers the implementation and validation of Lagrangian particle-based numerical methods — especially the Particle Finite Element Method (PFEM) and Moving Particle Semi-Implicit (MPS) method — for simulating non-Newtonian fluids governed by Herschel-Bulkley models and turbulent free-surface flows in marine environments. The works explore constitutive modeling, regularization techniques (e.g., Papanastasiou), and incorporation of turbulence closures such as constant eddy viscosity, mixing length, and k-ε models, focusing on convergence, stability, and accurate representation of free surfaces and wave dynamics in engineering applications.
Key finding: Implemented the Herschel-Bulkley constitutive law with Papanastasiou regularization within a Particle Finite Element Method framework in Kratos Multiphysics; validated the model against benchmark problems from literature... Read more
Key finding: Developed an MPS-based Lagrangian model incorporating three turbulence closures — constant eddy viscosity, mixing length, and k-ε model — to simulate wave evolution problems including run-up, wave train generation, and... Read more
Key finding: Evaluated the incorporation of turbulence models into MPS to improve hydrodynamic simulation of marine free surface flows; demonstrated that turbulence closures reduce numerical pressure oscillations and enhance stability and... Read more
Key finding: Renewed the MPS method by coupling it with turbulence models including mixing length and k-ε closures, enhanced spatial discretization with higher order MPS operators, and applied it to simulate nonlinear wave... Read more
5. What are the mathematical structures and solution relations for linear and implicit Lagrangian differential equations, and how do they interconnect?
This research area investigates the factorization of first-order linear and non-homogeneous differential operators and the relationships among solutions of three types of totally linear partial differential equations (homogeneous, non-homogeneous, and with source terms). By factorizing operators as products involving scalar functions and homogeneous operators, it becomes possible to transform between solution sets, simplifying integration and providing a structured framework for implicit Lagrangian differential systems.
Key finding: Presented a novel approach to solving first-order linear PDEs of Lagrange type via factorization of associated non-homogeneous differential operators into a product of a scalar function, a homogeneous operator, and its... Read more