Robert P. Lipton
Nicholson Professor Louisiana State University
Biography
Areas of Expertise
Research Focus
Wave Propagation & Metamaterials
Dr. Lipton’s research focuses on multiscale analysis of heterogeneous media, photonics and metamaterials, wave propagation, and fracture in advanced composites. He develops homogenization and nonlocal PDE frameworks to capture subwavelength resonances, dispersion, and damage evolution, informing transformational electromagnetics and the design of strong materials as LSU’s Nicholson Professor of Mathematics.
Education
NYU
Ph.D.
Mathematics
1986
NYU
M.S.
Mathematics
1984
University of Colorado
B.S.
Electrical Engineering
1981
Accomplishments
LSU Distinguished Research Master Award and University Medal
2021
Fellow, American Mathematical Society
2020
Fellow, Society for Industrial and Applied Mathematics
2020
Media Appearances
LSU Math Professor Robert Lipton Secures $1.25 Million MURI Award to Help Improve Materials Durability
Louisiana State University online
2024-04-03
"I'm interested in the dynamics of extreme deformation in material structures. And by extreme deformation, I mean ripping things apart and blowing things up," explains LSU Mathematics professor Robert Lipton. His research aims to enhance the understanding of damage propagation in heterogeneous materials.
Microwaving Saturn
WRKF 89.9 radio
2016-02-01
For most people, a microwave means a quick way to "nuke" your food.
But for LSU Math Professor Dr. Robert Lipton, a microwave means another thing: “Deep space communications – like how do you control the Mars Rover? They can use microwaves or radiowaves.”
Articles
Energy balance and damage for dynamic fast crack growth from a nonlocal formulation
Journal of Elasticity2025
A nonlocal model for dynamic brittle damage is introduced consisting of two phases, one elastic and the other inelastic. Evolution from the elastic to the inelastic phase depends on material strength. Existence and uniqueness of the displacement-failure set pair follow from an initial value problem describing the evolution. The displacement-failure pair satisfies energy balance. The length of nonlocality ϵ is taken to be small relative to the domain in R d, d= 2, 3. The strain is formulated as a difference quotient of the displacement in the nonlocal model. The two point force is expressed in terms of a weighted difference quotient and delivers an evolution on a subset of R d× R d. This evolution provides an energy balance between external energy, elastic energy, and damage energy including fracture energy.
Macroscopic effects of intraparticle fracture, grain topology and shape on vehicle dynamics and mobility over gravel road beds
Granular Matter2025
The hybrid particle-based computational platform that couples peridynamics with the discrete element method (PeriDEM) is used to model vehicle mobility over roadbeds. We consider wheels rolling over gravel beds, where gravel is allowed to deform and fracture. The motion of particles are not constrained to translation and rotation as in DEM and grains can deform elastically or inelastically. This allows for more modes of inter-particle interaction. The effects of gravel shape and topology on the vehicle mobility are examined using the higher fidelity modeling. Here we study how these aspects affect vehicle range, average vehicle velocity, traction as measured by wheel slip, as well as the overall energy needed to travel a prescribed distance.
Nodal finite element approximation of peridynamics
Computer Methods in Applied Mechanics and Engineering2025
This work considers the nodal finite element approximation of peridynamics, in which the nodal displacements satisfy the peridynamics equation at each mesh node. For the nonlinear bond-based peridynamics model, it is shown that, under the suitable assumptions on an exact solution, the discretized solution associated with the central-in-time and nodal finite element discretization converges to a solution in the L 2 norm at the rate C 1 Δ t+ C 2 h 2/ϵ 2. Here, Δ t, h, and ϵ are time step size, mesh size, and the size of the horizon or nonlocal length scale, respectively. Constants C 1 and C 2 are independent of h and Δ t and depend on norms of the solution and nonlocal length scale. Several numerical examples involving pre-crack, void, and notch are considered, and the efficacy of the proposed nodal finite element discretization is analyzed.
Fracture as an Emergent Phenomenon
Oberwolfach Reports2024
The mechanics of fracture propagation provides essential knowledge for the risk tolerance design of devices, structures, and vehicles. Techniques of free energy minimization provide guidance, but have limited applicability to material systems evolving away from equilibrium. Experimental evidence shows that the material response depends on driving forces arising from mechanical fields. Recent years have witnessed the development of new methods for modeling complex dynamic and quasistatic fracture. New approaches may differ remarkably from previous ones, as they involve implicit coupling between damaged and undamaged states, allowing fracture to be modeled as emergent phenomena.
A multiscale fracture model using peridynamic enrichment of finite elements within an adaptive partition of unity: Experimental validation
Mechanics Research Communications2024
Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Here, we apply Peridynamic (PD) enrichment to propagate cracks in the PUM global–local enrichment scheme. We apply linear elasticity globally and PD over local zones where fractures occur. The elastic fields provide appropriate boundary data for local PD simulations on a subdomain containing the crack tip to grow the crack. Once the updated crack path is found the elastic field in the body and surrounding the crack is updated using the PUM basis with an elastic field enrichment near the crack. The subdomain for the PD simulation is chosen to include the current crack tip as well as features that influence crack growth.
Affiliations
- American Association of the Advancement of Science (AAAS)
- Society for Engineering Science (SES)
- American Mathematical Society (AMS)
- Society for Industrial and Applied Mathematics (SIAM)
- Society for Natural Philosophy (SNP)
- Materials Research Society (MRS)
- United States Association for Computational Mechanics (USACM)
Event Appearances
Fracture as Emergent Phenomena
2024 | PDEs and Applied Mathematics “Celebration of Session 100th” Online
Dynamic Brittle Fracture as a Well Posed Nonlocal Initial Value Problem
2024 | Hausdorff Reasearch Institute for Mathematics Bonn, Germany
Multiscale Problems: Algorithms, Numerical Analysis and Computation
2024 | Hausdorff Institute Bonn, Germany
Research Grants
Complexity, Nonlocality, and Uncertainty in Heterogeneous Solids
ARO MURI
2024-2028
Title: Designing Mutable Metamaterials with Photo-Adaptive Meta-Atoms
NSF DMREF 1921707
2019-2024
Predicting and Controlling the response of Particulate Systems through Grain Scale Engineering
ARO MURI
2019-2025
Media
